Chapter 10 Visualising Solid Shapes (Additional Questions)

We need to identify the 3-dimensional shape among the given options.

Let's analyze each option:

(A) Circle: A circle is a closed curve in a plane. It is a 2-dimensional shape as it can be defined by points $(x, y)$ satisfying $x^2 + y^2 = r^2$ for a given radius $r$ in a 2D coordinate system.

(B) Triangle: A triangle is a polygon with three vertices, defined by three non-collinear points in a plane. It is a 2-dimensional shape.

(C) Square: A square is a quadrilateral with four equal sides and four right angles, defined in a plane. It is a 2-dimensional shape.

(D) Sphere: A sphere is the set of all points in three-dimensional space that are equally distant from a given point (the center). Its equation in a 3D coordinate system is $(x - x_0)^2 + (y - y_0)^2 + (z - z_0)^2 = r^2$, which clearly involves three dimensions. It is a 3-dimensional shape.

Comparing the properties of the given shapes, only the sphere exists in three dimensions.

Therefore, the correct option is (D) Sphere.

We are asked to identify the term for a shape that possesses length, width, and height.

Let's examine the characteristics of each option:

(A) 2-D shape: A 2-dimensional shape has only two dimensions, typically length and width. It exists on a plane.

(B) Plane figure: This is another term for a 2-dimensional shape. It lies entirely within a single plane.

(C) Solid shape: A solid shape, also known as a 3-dimensional shape, occupies space and has three dimensions: length, width, and height (or depth).

(D) Line segment: A line segment is a part of a line and has only one dimension, which is length.

Based on the definitions, a shape possessing length, width, and height fits the description of a solid shape or a 3-dimensional shape.

Therefore, the correct answer is (C) Solid shape.

We need to determine the number of faces on a cuboid.

Solution:

A cuboid is a three-dimensional solid bounded by six rectangular faces. It is a convex polyhedron.

Consider a typical cuboid, like a rectangular box or a brick.

It has:

1. A top face.

2. A bottom face (opposite to the top face).

3. A front face.

4. A back face (opposite to the front face).

5. A left side face.

6. A right side face (opposite to the left side face).

Counting these distinct surfaces, we find there are a total of 6 faces.

Thus, a cuboid has 6 faces.

The correct option is (B) 6.

We are asked to identify the term for the sharp points where three or more edges meet on a solid shape.

Let's define the key terms related to solid shapes:

Faces: These are the flat or curved surfaces that make up the boundary of a solid shape.

Edges: These are the line segments where two faces of a solid shape meet.

Vertices: These are the points where three or more edges of a solid shape meet. These points are often referred to as corners or sharp points.

Diagonals: These are line segments connecting two vertices that are not on the same edge or face (for space diagonals) or connecting two non-adjacent vertices on the same face (for face diagonals).

The description "sharp points where three or more edges meet" exactly defines a vertex.

Therefore, the correct answer is (C) Vertices.

We need to identify which of the given shapes is a polyhedron.

Solution:

A polyhedron is a three-dimensional solid whose boundary is composed of a finite number of flat polygonal faces. These faces meet at straight line segments called edges, and the edges meet at points called vertices. Key characteristics of a polyhedron are that it has only flat faces and straight edges, and there are no curved surfaces.

Let's examine each option:

(A) Cylinder: A cylinder has two flat circular bases and a curved lateral surface. Because it has a curved surface, it is not a polyhedron.

(B) Cone: A cone has one flat circular base and a curved lateral surface that tapers to a point (apex). Because it has a curved surface, it is not a polyhedron.

(C) Cube: A cube is a solid bounded by six square faces. All its faces are flat polygons, and all its edges are straight lines. Therefore, a cube is a polyhedron.

(D) Sphere: A sphere is a perfectly round geometrical object in three-dimensional space. Its surface is entirely curved. Because it has a curved surface, it is not a polyhedron.

Based on the definition, only the cube fits the description of a polyhedron.

The correct option is (C) Cube.

We are looking for the term that describes a solid shape where all its faces are polygons.

Let's consider the definitions of the given options:

(A) Circle: A circle is a 2-dimensional shape, not a solid shape.

(B) Polyhedron: A polyhedron (plural: polyhedra or polyhedrons) is a three-dimensional solid in space with flat polygonal faces, straight edges, and sharp corners or vertices. By definition, all faces of a polyhedron are polygons.

(C) Sphere: A sphere is a 3-dimensional shape, but its surface is entirely curved. It does not have polygonal faces.

(D) Curved solid: This is a general term for a solid shape that has at least one curved surface. Examples include spheres, cylinders, and cones. These shapes do not have all faces as polygons.

Based on the definitions, the term for a solid shape whose faces are all polygons is a polyhedron.

The correct answer is (B) Polyhedron.

We need to find the number of edges in a triangular prism.

Solution:

A triangular prism is a polyhedron with two parallel and congruent triangular bases and three rectangular faces connecting the corresponding sides of the bases.

Let's count the edges:

1. Edges on the top triangular base: There are 3 edges forming the triangle.

2. Edges on the bottom triangular base: There are 3 edges forming the triangle, parallel to the top base edges.

3. Edges connecting the vertices of the top base to the corresponding vertices of the bottom base (lateral edges): There are 3 such edges.

Total number of edges = (Edges on top base) + (Edges on bottom base) + (Lateral edges)

Total number of edges = $3 + 3 + 3 = 9$.

Thus, a triangular prism has 9 edges.

The correct option is (B) 9.

We need to determine the shape observed when a cylinder is viewed from the front.

Solution:

Imagine a cylinder standing upright or lying on its side. The "front view" usually refers to looking at the cylinder from a direction perpendicular to its axis, such that you see its profile.

When you look at a cylinder from the side, you see the rectangular shape of its lateral surface. The height of this rectangle corresponds to the height of the cylinder, and the width corresponds to the diameter of its circular base.

Let's consider the views:

Top view (looking down from above): A circle.

Side view (looking from the front): A rectangle (or a square if the height equals the diameter).

Bottom view (looking up from below): A circle.

The front view is typically considered a side view, showing the profile along its height and diameter.

Therefore, the front view of a cylinder is a Rectangle.

The correct option is (B) Rectangle.

We need to determine the shape observed when a cone is viewed from the top.

Solution:

Imagine a cone standing upright, with its circular base on a surface and its apex pointing upwards.

When you look down directly from above the cone, you are essentially looking at its circular base.

However, the apex (the sharp point at the top) is located in the very center of the circular base from this perspective.

So, the view from the top shows the circular base with the apex appearing as a point exactly in the middle of the circle.

Therefore, the top view of a cone is usually a Point inside a circle.

The correct option is (C) Point inside a circle.

We need to determine the shape observed when a cube is viewed from the top, front, or side.

Solution:

A cube is a three-dimensional solid bounded by six equal square faces.

Consider a standard cube. Each face of the cube is a perfect square.

When you look at the cube directly from the front, you are looking at one of its faces. This face is a square.

When you look at the cube directly from the top, you are looking at the top face. This face is also a square.

When you look at the cube directly from the side, you are looking at one of its side faces. This face is also a square.

In each of these standard orthogonal views (top, front, or side), the visible projection of the cube onto a plane parallel to the face being viewed is a square.

Therefore, the view of a cube from the top, front, or side is always a Square.

The correct option is (B) Square.

We need to identify the statement that is NOT true regarding the views of a standard sphere.

Solution:

A sphere is a perfectly symmetrical three-dimensional object. Its shape is the same from every direction.

Let's consider the standard orthogonal views:

(A) Top view: When viewed from directly above, the projection of a sphere onto a plane is a circle. This statement is True.

(B) Front view: When viewed from the front, the projection of a sphere onto a plane is a circle. This statement is True.

(C) Side view: When viewed from the side, the projection of a sphere onto a plane is a circle. This statement is True.

Since the top, front, and side views (and indeed any cross-sectional view through the center) of a standard sphere are always circles of the same diameter, the views are not different; they are all identical circles.

(D) All views are different: As established above, the standard views (top, front, side) are all circles and are the same shape. Therefore, this statement is False.

The statement that is NOT a view or property of the view of a standard sphere is that all views are different.

The correct option is (D) All views are different.

We are given Euler's formula for polyhedra and the number of faces and vertices for a specific polyhedron. We need to find the number of edges.

Given:

Euler's formula: $F + V - E = 2$

Number of faces, $F = 7$

Number of vertices, $V = 10$

To Find:

Number of edges, $E$

Solution:

We use Euler's formula:

Substitute the given values of $F$ and $V$ into the formula:

Simplify the equation:

To find $E$, we rearrange the equation:

The polyhedron has 15 edges.

The correct option is (A) 15.

We need to determine if a polyhedron can exist with the given number of faces, vertices, and edges by checking if these numbers satisfy Euler's formula.

Given:

Number of faces, $F = 10$

Number of vertices, $V = 15$

Number of edges, $E = 23$

Verification using Euler's Formula:

Euler's formula for any simple polyhedron states that:

Substitute the given values of $F$, $V$, and $E$ into the left-hand side of the formula:

Calculate the value:

The result of $F + V - E$ is 2, which is equal to the right-hand side of Euler's formula.

Since the given numbers satisfy Euler's formula ($10 + 15 - 23 = 2$), it is possible for a polyhedron to have these characteristics.

Therefore, the answer is Yes.

The correct option is (A) Yes.

We need to find a solid shape that satisfies the following conditions:

1. Top view is different from the front view.

2. Side view is the same as the front view.

Let's examine the views of each option:

(A) Cube: A cube has equal length, width, and height. The top view is a square, the front view is a square, and the side view is a square. All three standard views are the same (congruent squares). This does not satisfy condition 1.

(B) Cylinder: Consider a cylinder standing upright or lying on its side. If we take the "front view" and "side view" to be orthogonal projections perpendicular to the axis, and the "top view" to be parallel to the axis:

- The top view (looking down on a standing cylinder) is a circle.

- The front view (looking from the side of a standing cylinder) is a rectangle.

- The side view (looking from another side of a standing cylinder) is the same rectangle as the front view.

The top view (circle) is different from the front view (rectangle). The side view (rectangle) is the same as the front view (rectangle). This fits both conditions.

(C) Cuboid (with unequal length, width, height): Let the dimensions be length $l$, width $w$, and height $h$, where $l \neq w$, $w \neq h$, and $l \neq h$.

- Front view (e.g., face with dimensions $l \times h$) is a rectangle of size $l \times h$.

- Side view (e.g., face with dimensions $w \times h$) is a rectangle of size $w \times h$.

- Top view (e.g., face with dimensions $l \times w$) is a rectangle of size $l \times w$.

Since $l \neq w$, the side view ($w \times h$) will generally be different from the front view ($l \times h$) (unless $h=0$, which isn't a solid). If the dimensions are strictly unequal, none of the views are the same. If, however, two dimensions were equal (e.g., $l=w \neq h$), then the front view ($l \times h$) and side view ($l \times h$) would be the same, while the top view ($l \times l$) would be different. But the question specifies "unequal length, width, height", implying all three are different.

(D) Sphere: A sphere is perfectly symmetrical. The top view, front view, and side view are all congruent circles. All three standard views are the same. This does not satisfy condition 1.

Based on the analysis, the cylinder consistently satisfies the given conditions regardless of its dimensions (as long as it has non-zero height and radius), while the cuboid with strictly unequal dimensions does not. A cuboid only fits the description if two dimensions are equal and the third is different, which contradicts the specific type of cuboid mentioned in option (C).

Therefore, the object is a Cylinder.

The correct option is (B) Cylinder.

We need to match each solid shape with its corresponding view characteristics.

Let's analyse each solid shape:

(i) Rectangular Prism: A rectangular prism has a rectangular base and its front face is a rectangle (assuming standard orientation where the front is one of the rectangular sides). Thus, its front view is a rectangle and its base is a rectangle. This matches option (b).

(ii) Triangular Pyramid: A triangular pyramid has a triangular base and its faces are triangles. When viewed from the front (looking at one of the triangular faces), the view is a triangle. Its base is a triangle. This matches option (c).

(iii) Cone: A cone has a circular base and its lateral surface is curved. When viewed from the front or side, the outline is a triangle. Its base is a circle. This matches option (d).

(iv) Square Pyramid: A square pyramid has a square base and its faces are triangles. When viewed from the front (looking at one of the triangular faces), the view is a triangle. Its base is a square. This matches option (a).

The correct matches are:

(i) - (b)

(ii) - (c)

(iii) - (d)

(iv) - (a)

Let's check the given options to find the correct combination:

(A) (i)-(b), (ii)-(c), (iii)-(d), (iv)-(a) - This matches our findings.

(B) (i)-(b), (ii)-(a), (iii)-(d), (iv)-(c) - Incorrect.

(C) (i)-(a), (ii)-(c), (iii)-(d), (iv)-(b) - Incorrect.

(D) (i)-(c), (ii)-(b), (iii)-(a), (iv)-(d) - Incorrect.

The correct option is (A).

We need to identify the solid shape from the options that has exactly one vertex.

Let's count the vertices for each solid shape:

(A) Cuboid: A cuboid has 8 vertices (4 on the top face and 4 on the bottom face).

(B) Cylinder: A cylinder has no vertices. It has two circular edges, but no points where edges meet.

(C) Cone: A cone has a circular base and its lateral surface tapers to a single point called the apex. This apex is the only vertex of the cone. So, a cone has 1 vertex.

(D) Prism: The number of vertices of a prism depends on the shape of its base. A prism with an n-sided base has $2n$ vertices (n vertices on the top base and n vertices on the bottom base). For example, a triangular prism has 6 vertices ($2 \times 3$), and a rectangular prism has 8 vertices ($2 \times 4$). A prism always has more than one vertex (unless n=1, which is not a standard prism).

Based on the vertex count, only the cone has exactly one vertex.

Therefore, the correct answer is (C) Cone.

We need to evaluate the truthfulness of the given Assertion (A) and Reason (R) and determine the relationship between them.

Assertion (A): A sphere is a polyhedron.

A polyhedron is defined as a three-dimensional solid whose boundary is composed of a finite number of flat polygonal faces. A sphere, on the other hand, is a solid bounded by a single curved surface.

Since a sphere does not have flat polygonal faces, it is not a polyhedron.

Therefore, Assertion (A) is False.

Reason (R): A polyhedron is a solid bounded by polygons.

This statement is consistent with the definition of a polyhedron, where the boundary surfaces (faces) are polygons.

Therefore, Reason (R) is True.

Now we compare our findings with the given options:

- Option (A) states both are true, which is incorrect as A is false.

- Option (B) states both are true, which is incorrect as A is false.

- Option (C) states A is true and R is false, which is incorrect as A is false and R is true.

- Option (D) states A is false and R is true, which matches our findings.

Thus, Assertion (A) is false, and Reason (R) is true.

The correct option is (D) A is false, but R is true.

We need to complete the sentence by identifying the correct term for the flat surfaces of a solid shape.

Let's consider the definitions of the components of a solid shape, particularly a polyhedron which has flat surfaces:

Edges: These are the line segments where two flat surfaces meet.

Vertices: These are the points where three or more edges meet.

Faces: These are the flat or curved surfaces that form the boundary of a solid shape. For polyhedra, the faces are flat and polygonal.

Sides: While sometimes used informally, 'sides' is less precise than 'faces' for the bounding surfaces of a 3D object.

The term that specifically refers to the flat surfaces of a solid shape is faces.

Completing the sentence: The flat surfaces of a solid shape are called its Faces.

The correct option is (C) Faces.

We need to identify the solid shape from the options that has neither vertices nor edges.

Let's examine the properties of each solid shape in terms of vertices and edges:

(A) Cone: A cone has one vertex (the apex) and one edge (the circular base). It does not fit the description of having no vertex and no edge.

(B) Cylinder: A cylinder has no vertices, but it has two circular edges (the boundaries of the top and bottom bases). It does not fit the description of having no vertex and no edge.

(C) Sphere: A sphere is a perfectly round three-dimensional object. Its boundary is a single continuous curved surface. It has no points where edges meet (no vertices) and no line segments where faces meet (no edges). It fits the description of having no vertex and no edge.

(D) Cube: A cube is a polyhedron. It has 8 vertices and 12 edges. It does not fit the description of having no vertex and no edge.

Based on the analysis, the solid shape that has no vertex and no edge is the sphere.

Therefore, the correct answer is (C) Sphere.

We are given a case study describing a building design that incorporates several solid shapes: a rectangular prism, a hemisphere (dome), and cylinders. We need to identify which of the options represents a polyhedron used in this design.

Solution:

First, let's recall the definition of a polyhedron.

A polyhedron is a three-dimensional solid whose boundary is made up entirely of flat polygonal faces. Polyhedra have no curved surfaces.

Now let's examine each of the shapes mentioned in the case study and listed in the options:

Rectangular Prism: A rectangular prism is a solid bounded by six rectangular faces. Since rectangles are polygons and all faces are flat, a rectangular prism fits the definition of a polyhedron.

Hemisphere: A hemisphere is half of a sphere. A sphere is bounded by a curved surface. Therefore, a hemisphere also has a curved surface and is not a polyhedron.

Cylinder: A cylinder has two flat circular bases and a curved lateral surface. Because it has a curved surface, it is not a polyhedron.

Dome: The case study states the dome is shaped like a hemisphere. As discussed, a hemisphere has a curved surface and is not a polyhedron.

Among the shapes mentioned in the design, only the rectangular prism is a polyhedron because it is bounded solely by flat polygonal faces.

Therefore, the polyhedron used in the design is the Rectangular prism.

The correct option is (A) Rectangular prism.

We need to identify the solid shape(s) from the given options that have a curved surface, based on the case study in Question 20.

The case study mentions the following shapes used in the design:

- A rectangular prism (main block)

- A hemisphere (dome)

- Cylindrical columns

Let's examine each of the options:

(A) Rectangular prism: A rectangular prism is a polyhedron bounded by six flat rectangular faces. It has no curved surfaces.

(B) Cylinder: A cylinder has two flat circular bases and a curved lateral surface. Therefore, a cylinder has a curved surface.

(C) Hemisphere: A hemisphere is half of a sphere. The surface of a sphere is entirely curved. Therefore, a hemisphere has a curved surface.

(D) Both (B) and (C): This option suggests that both cylinders and hemispheres have curved surfaces.

Based on our analysis, both the cylinder and the hemisphere have curved surfaces. Option (D) correctly includes both of these shapes.

Therefore, the correct answer is (D) Both (B) and (C).

We need to determine the number of faces on a triangular pyramid.

Solution:

A triangular pyramid is a polyhedron with a triangular base and three triangular faces that meet at a point called the apex.

Let's count the faces:

1. The base: It is a triangle, so there is 1 base face.

2. The lateral faces: There are three triangular faces connecting the sides of the base to the apex. So there are 3 lateral faces.

Total number of faces = (Base faces) + (Lateral faces)

Total number of faces = $1 + 3 = 4$

Thus, a triangular pyramid has 4 faces.

The correct option is (B) 4.

We need to identify the solid shape that has a circular base and a single vertex at the top.

Let's consider the characteristics of each shape:

(A) Cylinder: A cylinder has two parallel circular bases and a curved lateral surface. It has no vertices.

(B) Sphere: A sphere is a perfectly round solid with a single curved surface. It has no base and no vertices.

(C) Cone: A cone has a circular base and a curved lateral surface that tapers to a single point called the apex or vertex. This fits the description of having a circular base and a vertex at the top (apex).

(D) Pyramid: A pyramid has a polygonal base (e.g., square, triangular, etc.) and triangular faces that meet at a single vertex (apex). While it has a vertex at the top, its base is polygonal, not necessarily circular.

The shape that specifically has a circular base and one vertex at the top is a cone.

Therefore, the correct answer is (C) Cone.

We need to identify which of the given solid shapes is a regular polyhedron.

Definition of a Regular Polyhedron:

A regular polyhedron is a polyhedron whose faces are congruent regular polygons and where the same number of faces meet at each vertex. There are only five convex regular polyhedra, known as the Platonic solids: tetrahedron, cube, octahedron, dodecahedron, and icosahedron.

Let's examine each option:

(A) Cube: A cube has 6 faces, and each face is a square (a regular polygon). At each vertex, exactly 3 square faces meet. All faces are congruent, and the same number of faces meet at each vertex. Therefore, a cube is a regular polyhedron (it is one of the Platonic solids).

(B) Rectangular prism: A rectangular prism has rectangular faces. If the rectangle is not a square (i.e., length and width are unequal), the faces are not regular polygons. Even if the faces are squares (making it a cube), a general rectangular prism does not necessarily have all faces as congruent regular polygons. At vertices where faces of different dimensions meet, the vertex configuration is not the same throughout unless it is a cube.

(C) Triangular prism: A triangular prism has two triangular bases and three rectangular lateral faces. For it to be regular, the bases would need to be equilateral triangles, and the lateral faces would need to be congruent rectangles (possibly squares). However, not all faces are the same type of regular polygon (triangles and rectangles/squares). Thus, it is not a regular polyhedron in the strict sense.

(D) Square pyramid: A square pyramid has a square base and four triangular lateral faces. For it to be regular, the base is a square (regular polygon), and the lateral faces are congruent isosceles triangles. However, the faces are not all the same type of regular polygon (squares and triangles), so it is not a regular polyhedron.

Among the given options, only the cube satisfies the conditions of being a regular polyhedron.

Therefore, the correct answer is (A) Cube.

We need to determine the shape observed when a traffic cone kept upright is viewed from the top.

Solution:

A traffic cone is essentially a cone shape. When it is kept upright, its circular base is at the bottom, and its apex (the sharp point) is at the top.

The top view means looking down on the cone from directly above its apex.

From this perspective, you see the entire circular base.

The apex, being directly above the center of the base, appears as a single point exactly at the center of the observed circle.

Therefore, the top view of an upright traffic cone is a Circle with a point at the center.

Let's consider the other options:

(B) Triangle: A triangle is the front or side view of a cone.

(C) Circle: While the base is a circle, the apex is visible at the center in the top view of an upright cone, making option (A) more precise.

(D) Square: A square is not a typical view of a cone.

The correct option is (A) Circle with a point at the center.

We need to find the pair with the same number of faces.

Let's list the number of faces for each shape:

Cube: 6 faces

Rectangular prism: 6 faces

Triangular prism: 5 faces

Square pyramid: 5 faces

Triangular pyramid: 4 faces

Now compare the pairs in the options:

(A) Cube (6) and Rectangular prism (6). Number of faces is the same ($6=6$).

(B) Triangular prism (5) and Square pyramid (5). Number of faces is the same ($5=5$).

(C) Cube (6) and Square pyramid (5). Number of faces is different ($6 \neq 5$).

(D) Triangular prism (5) and Triangular pyramid (4). Number of faces is different ($5 \neq 4$).

Both options (A) and (B) contain pairs with the same number of faces. Assuming a single correct answer is expected, we choose option (A) as both shapes are types of cuboids.

The correct option is (A).

We need to evaluate the truthfulness of Assertion (A) and Reason (R) and determine if R correctly explains A.

Assertion (A): A prism is a polyhedron.

A polyhedron is a three-dimensional solid whose boundary is composed entirely of flat polygonal faces. A prism is defined by its polygonal bases and parallelogram (or rectangular) lateral faces. Since all the faces of a prism are polygons and are flat, a prism fits the definition of a polyhedron.

Therefore, Assertion (A) is True.

Reason (R): Its bases are congruent polygons and its side faces are parallelograms.

This statement describes the geometric properties of a prism. A prism does indeed have two parallel and congruent polygonal bases, and its lateral faces connecting these bases are parallelograms.

Therefore, Reason (R) is True.

Now let's check if Reason (R) is the correct explanation for Assertion (A).

The reason why a prism is a polyhedron is precisely because its boundary is made up of flat polygonal faces (the bases and the lateral faces which are parallelograms). Reason (R) describes these polygonal faces (bases and lateral faces), which are the defining characteristics that classify a prism as a polyhedron.

Thus, Reason (R) correctly explains why Assertion (A) is true.

Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct explanation of Assertion (A).

The correct option is (A) Both A and R are true, and R is the correct explanation of A.

We need to complete the sentence by identifying the correct term for the line segments that form the skeleton of a solid shape.

Let's consider the common parts of a solid shape:

Faces: These are the flat (or curved) surfaces that make up the outside of the shape.

Vertices: These are the points or corners where multiple edges meet.

Edges: These are the line segments where two faces meet. They form the structural outline or "skeleton" of the shape.

Corners: This term is often used informally for vertices.

The term that refers to the line segments forming the skeleton of a solid shape is edges.

Completing the sentence: The line segments that form the skeleton of a solid shape are called its Edges.

The correct option is (C) Edges.

We are given information about a rectangular prism box and need to identify the statement that is always true for any rectangular prism.

A rectangular prism is a solid shape with 6 faces, 12 edges, and 8 vertices. Its faces are rectangles.

Let's evaluate each statement:

(A) All its faces are squares. This is only true if the rectangular prism is a cube (i.e., when $l=w=h$). A general rectangular prism has rectangular faces, which may or may not be squares. This is not always true.

(B) It has 12 edges. A rectangular prism always has 12 edges, where the faces meet. There are 4 edges of length $l$, 4 edges of width $w$, and 4 edges of height $h$, summing up to 12 edges. This is always true for any rectangular prism.

(C) It has 6 square faces. A rectangular prism has 6 faces, but they are rectangular, not necessarily square. This is only true if the prism is a cube. This is not always true.

(D) The length, width, and height are always equal. This condition defines a cube ($l=w=h$). A rectangular prism can have different values for length, width, and height. This is not always true.

The only statement that holds true for any rectangular prism is that it has 12 edges.

The correct option is (B) It has 12 edges.

We are given that the cubical box is a cube and its side length is $s = 10 \text{ cm}$. We need to find the number of vertices on a cube.

Solution:

A cube is a specific type of rectangular prism where all edges are of equal length. It is a polyhedron with 6 square faces and 12 edges.

The vertices of a solid shape are the points where the edges meet.

For a cube, we can count the vertices:

- There are 4 vertices on the top face.

- There are 4 vertices on the bottom face.

Total number of vertices = $4 + 4 = 8$.

Alternatively, using Euler's formula ($F + V - E = 2$) for a cube:

Number of faces ($F$) = 6

Number of edges ($E$) = 12

Substituting into the formula: $6 + V - 12 = 2$

Simplifying: $V - 6 = 2$

Solving for $V$: $V = 2 + 6 = 8$

The side length $s = 10 \text{ cm}$ is given, but it does not affect the number of vertices, which is a property of the shape itself, not its size.

A cubical box always has 8 vertices.

The correct option is (B) 8.

We are given the dimensions of a rectangular prism box as $5 \text{ cm} \times 8 \text{ cm} \times 12 \text{ cm}$. We need to find the number of edges on a rectangular prism.

Solution:

A rectangular prism is a polyhedron with 6 rectangular faces, 12 edges, and 8 vertices.

The dimensions ($l=12 \text{ cm}$, $w=8 \text{ cm}$, $h=5 \text{ cm}$ or any permutation thereof) define the size of the prism, but they do not change the fundamental number of edges the shape possesses.

In a rectangular prism, there are sets of parallel edges of equal length:

- 4 edges corresponding to the length ($12 \text{ cm}$).

- 4 edges corresponding to the width ($8 \text{ cm}$).

- 4 edges corresponding to the height ($5 \text{ cm}$).

Total number of edges = $4 + 4 + 4 = 12$.

The number of edges in a rectangular prism is always 12, regardless of its specific dimensions, as long as they are non-zero.

The correct option is (B) 12.

We need to identify which of the given objects are examples of polyhedra.

Recall that a polyhedron is a three-dimensional solid whose boundary is composed entirely of flat polygonal faces. This means a polyhedron cannot have any curved surfaces.

Let's examine each option:

(A) A standard dice: A standard dice is shaped like a cube. A cube is a polyhedron because it is bounded by 6 flat square faces. So, a standard dice is an example of a polyhedron.

(B) A book: A book is typically shaped like a rectangular prism. A rectangular prism is a polyhedron because it is bounded by 6 flat rectangular faces. So, a book is an example of a polyhedron.

(C) A tennis ball: A tennis ball is shaped like a sphere. A sphere is bounded by a curved surface and has no flat polygonal faces. So, a tennis ball is not a polyhedron.

(D) A Rubik's cube: A Rubik's cube is a puzzle based on a cube shape. The overall shape is a cube, which is a polyhedron. The smaller cubes that make up the Rubik's cube are also polyhedra. So, a Rubik's cube is an example of a polyhedron.

(E) A water bottle: A typical water bottle usually has a cylindrical part and often a curved top or base. The presence of curved surfaces means it is not a polyhedron.

Based on the definition of a polyhedron, the objects that are polyhedra from the list are a standard dice, a book, and a Rubik's cube.

Therefore, the correct options are (A), (B), and (D).

We need to identify the statement that is FALSE regarding a triangular pyramid.

Let's analyze the properties of a triangular pyramid:

A triangular pyramid (also known as a tetrahedron) has a triangular base and three triangular lateral faces that meet at a common vertex (apex).

Let's evaluate each statement:

(A) It has 4 faces. A triangular pyramid has 1 base face (a triangle) and 3 lateral faces (triangles). Total faces = $1 + 3 = 4$. This statement is True.

(B) It has 4 vertices. A triangular base has 3 vertices. The three lateral faces meet at one additional vertex (the apex). Total vertices = $3 + 1 = 4$. This statement is True.

(C) Its base is a triangle. By definition, a triangular pyramid has a triangular base. This statement is True.

(D) All its faces are congruent triangles. A triangular pyramid has 4 triangular faces. While the three lateral faces might be congruent to each other if the base is equilateral and the apex is directly above the centroid, the base triangle itself is generally not congruent to the lateral triangles. A triangular pyramid where all four faces are congruent equilateral triangles is called a regular tetrahedron. However, a general triangular pyramid does not require all its faces to be congruent. This statement is False for a general triangular pyramid.

The statement that is FALSE about a triangular pyramid is that all its faces are congruent triangles.

The correct option is (D) All its faces are congruent triangles.

We are asked about the shape of the cross-section when a cylindrical log is sliced vertically.

Solution:

Imagine a cylindrical log standing upright. A "vertical slice" means cutting downwards, perpendicular to the base, along the length of the cylinder.

If the vertical slice passes through the center of the circular bases, the resulting cross-section will be a rectangle whose height is the height of the cylinder and whose width is the diameter of the circular base.

If the vertical slice does not pass through the center (i.e., it's offset), the resulting cross-section will still be a rectangle, but its width will be less than the diameter of the base.

In either case, a vertical slice of a cylinder results in a rectangular cross-section.

Let's consider the other options:

(A) Circle: A circular cross-section is obtained by slicing the cylinder horizontally, parallel to the base.

(C) Oval: An oval (ellipse) cross-section is obtained by slicing the cylinder at an angle, not parallel or perpendicular to the base or height.

(D) Triangle: A triangle is a cross-section of shapes like cones or pyramids, not cylinders.

Therefore, if you slice a cylindrical log vertically, the cross-section is a Rectangle.

The correct option is (B) Rectangle.

We are given a relationship between the number of faces ($F$), vertices ($V$), and edges ($E$) of a polyhedron, which is $F + V - E = 2$. We need to identify the name of this formula.

Let's consider the given options:

(A) Pythagoras theorem: This theorem relates the sides of a right-angled triangle, stating that the square of the hypotenuse is equal to the sum of the squares of the other two sides ($a^2 + b^2 = c^2$). This is unrelated to the faces, vertices, and edges of a polyhedron.

(B) Euler's formula: For any convex polyhedron, the relationship between the number of faces ($F$), vertices ($V$), and edges ($E$) is given by the formula $F + V - E = 2$. This formula is known as Euler's formula (specifically, Euler's polyhedron formula).

(C) Area formula: Area formulas calculate the amount of space occupied by a two-dimensional shape or surface. They are not related to the count of faces, vertices, and edges in the manner described.

(D) Perimeter formula: Perimeter formulas calculate the distance around the boundary of a two-dimensional shape. They are not related to the properties of three-dimensional solids in this way.

The formula $F + V - E = 2$ for polyhedra is a fundamental concept in topology and geometry and is correctly identified as Euler's formula.

Therefore, the correct answer is (B) Euler's formula.

We are given the number of faces ($F=5$), vertices ($V=6$), and edges ($E=9$) of a solid shape and need to identify the shape from the options.

Let's check the properties ($F$, $V$, $E$) of each solid shape in the options:

(A) Square pyramid:

Faces ($F$): 1 square base + 4 triangular lateral faces = 5

Vertices ($V$): 4 on the base + 1 apex = 5

Edges ($E$): 4 on the base + 4 lateral edges = 8

Properties: $F=5, V=5, E=8$. This does not match the given numbers.

(B) Triangular pyramid:

Faces ($F$): 1 triangular base + 3 triangular lateral faces = 4

Vertices ($V$): 3 on the base + 1 apex = 4

Edges ($E$): 3 on the base + 3 lateral edges = 6

Properties: $F=4, V=4, E=6$. This does not match the given numbers.

(C) Triangular prism:

Faces ($F$): 2 triangular bases + 3 rectangular lateral faces = 5

Vertices ($V$): 3 on the top base + 3 on the bottom base = 6

Edges ($E$): 3 on the top base + 3 on the bottom base + 3 lateral edges = 9

Properties: $F=5, V=6, E=9$. This matches the given numbers.

(D) Rectangular pyramid:

Faces ($F$): 1 rectangular base + 4 triangular lateral faces = 5

Vertices ($V$): 4 on the base + 1 apex = 5

Edges ($E$): 4 on the base + 4 lateral edges = 8

Properties: $F=5, V=5, E=8$. This does not match the given numbers.

The shape that has 5 faces, 6 vertices, and 9 edges is a triangular prism.

We can also verify with Euler's formula: $F + V - E = 5 + 6 - 9 = 11 - 9 = 2$. The numbers satisfy Euler's formula for a polyhedron.

The correct option is (C) Triangular prism.

We are asked for the shape of the side view of a triangular prism that is standing on its triangular base.

Solution:

Imagine a triangular prism placed on a surface such that its triangular bases are horizontal (one at the bottom, one at the top). The lateral faces are vertical rectangles (or parallelograms, but often rectangular in a right prism).

The "side view" means looking at the prism from a direction perpendicular to one of its lateral rectangular faces.

When viewed from the side, you see one of the rectangular lateral faces directly.

The height of this rectangle is the height of the prism, and the width is the length of one of the sides of the triangular base.

Let's consider the views of a triangular prism standing on its base:

Top view (looking down): A triangle.

Front view (looking at a lateral face): A rectangle.

Side view (looking at an adjacent lateral face): A rectangle.

Note: The front and side views will be rectangles. If the triangular base is equilateral and the lateral faces are squares, then the side view would be a square, which is a specific type of rectangle. However, the most general description for the side view is a rectangle.

Therefore, the side view of a triangular prism standing on its triangular base is a Rectangle.

The correct option is (B) Rectangle.

We need to find the number of vertices on a pyramid with a hexagonal base.

Solution:

A pyramid consists of a polygonal base and triangular lateral faces that meet at a single point called the apex.

The vertices of a pyramid are the vertices of its base plus the apex.

The base of this pyramid is a hexagon.

1. Number of vertices on the hexagonal base: A hexagon is a polygon with 6 vertices.

2. Number of vertices at the apex: There is 1 apex where all the lateral faces meet.

Total number of vertices = (Vertices on the base) + (Apex)

Total number of vertices = $6 + 1 = 7$.

A pyramid with an n-sided base has $n + 1$ vertices.

In this case, the base is a hexagon, so $n=6$.

Number of vertices = $6 + 1 = 7$.

Therefore, a pyramid with a hexagonal base has 7 vertices.

The correct option is (B) 7.

Differentiation between a 2-dimensional shape and a 3-dimensional shape:

A 2-dimensional (2D) shape is a shape that has only two dimensions: length and width (or breadth). It lies completely flat on a plane. 2D shapes have area but no volume.

Examples of common 2D shapes include squares, circles, triangles, rectangles, etc.

A 3-dimensional (3D) shape is a shape that has three dimensions: length, width, and height (or depth). These shapes occupy space and are not flat. 3D shapes have both surface area and volume.

Examples of common 3D shapes include cubes, spheres, cones, cylinders, pyramids, etc.

The key difference lies in the number of dimensions they possess: 2D shapes have two, while 3D shapes have three.

Examples from surroundings:

An example of a 2-dimensional shape from our surroundings is the surface of a sheet of paper or the face of a clock.

An example of a 3-dimensional shape from our surroundings is a book, a ball, or a glass.

(a) Square Pyramid

A square pyramid has a square base and four triangular faces that meet at a single point (apex).

Number of Faces: A square pyramid has 5 faces (1 square base + 4 triangular sides).

Number of Edges: A square pyramid has 8 edges (4 edges on the square base + 4 edges connecting the base vertices to the apex).

Number of Vertices: A square pyramid has 5 vertices (4 vertices on the square base + 1 apex vertex).

(b) Triangular Prism

A triangular prism has two parallel triangular bases and three rectangular side faces connecting the corresponding sides of the two bases.

Number of Faces: A triangular prism has 5 faces (2 triangular bases + 3 rectangular sides).

Number of Edges: A triangular prism has 9 edges (3 edges on each triangular base + 3 edges connecting the two bases).

Number of Vertices: A triangular prism has 6 vertices (3 vertices on each triangular base).

The shape of the faces of a triangular pyramid is a triangle.

A triangular pyramid has a triangular base and three triangular side faces that meet at an apex.

It has a total of 4 faces.

The solid shape that fits the description of having 6 faces, 12 edges, and 8 vertices is a Cuboid (or a Cube).

Given:

Number of Faces (F) = 6

Number of Edges (E) = 12

Number of Vertices (V) = 8

To Verify: Euler's formula, which states $F + V - E = 2$.

Verification:

Substitute the given values into Euler's formula:

Since $F + V - E = 2$, Euler's formula is verified for a cuboid/cube with the given number of faces, edges, and vertices.

A polyhedron is a three-dimensional solid shape that is bounded by flat surfaces called faces. These faces are polygons. The edges of a polyhedron are the line segments where two faces meet, and the vertices are the points where three or more edges meet.

No, a cylinder is not a polyhedron.

The reason a cylinder is not a polyhedron is that its surface includes curved faces (the top and bottom circular bases are flat, but the side surface connecting them is curved). According to the definition of a polyhedron, all its faces must be flat polygons. Since a cylinder has a curved surface, it does not meet this requirement.

A net of a 3D shape is a 2-dimensional (2D) pattern that, when folded along its edges, forms the 3-dimensional (3D) shape. It shows all the faces of the 3D shape laid out flat in a single plane.

Rough Sketch of the Net of a Cuboid:

A cuboid has 6 rectangular faces. There are several possible nets for a cuboid. A common net consists of four rectangular faces arranged in a row (representing the sides), with one base attached to one of the rectangles and the other base attached to the opposite rectangle in the row.

Imagine the rectangles are labelled F1, F2, F3, F4 (the sides) and B1, B2 (the bases).

A typical net layout could be represented conceptually like this (top view when flat):

          +-----+
          |  B1 |
    +-----+-----+-----+-----+
    |  F1 |  F2 |  F3 |  F4 |
    +-----+-----+-----+-----+
          |  B2 |
          +-----+
        

Here, B1 and B2 are the top and bottom bases, and F1, F2, F3, F4 are the side faces. When folded, F1 and F3 would be opposite sides, and F2 and F4 would be the other pair of opposite sides, with B1 and B2 forming the top and bottom.

A net of a cone is a 2D pattern that, when folded, forms a 3D cone. It consists of two parts: the base and the lateral (curved) surface.

Rough Sketch of the Net of a Cone:

The net of a cone is typically represented as a circle (for the base) and a sector of a larger circle (for the lateral surface).

             /\
            /  \
           /    \
          /      \
         /        \
        /          \
       /            \
      /              \
     /                \
    /__________________\
        (Sector of a Circle)

           O
          / \
         /   \
        /     \
       /       \
      -----------
      (Base Circle)
        

(Note: The arc length of the sector is equal to the circumference of the base circle).

Shapes involved in the net:

The shapes involved in the net of a cone are:

1. A Circle (representing the base)

2. A Sector of a Circle (representing the curved lateral surface)

The 'views' of a 3D shape are the 2-dimensional (2D) representations of the shape as seen from different angles or directions. Since a 3D shape occupies space and cannot be fully represented on a flat surface (like paper or a screen) in one image, looking at it from specific viewpoints helps to understand its structure, proportions, and details.

Essentially, a view shows what you would see if you were looking at the 3D object directly from that particular direction, projecting it onto a flat plane.

The common views we typically consider for a 3D shape are:

1. Front View (also called Elevation): What you see when looking directly at the front of the object.

2. Side View (often Left Side View or Right Side View): What you see when looking directly at one of the sides of the object.

3. Top View (also called Plan View): What you see when looking directly down at the object from above.

These different views, when presented together, provide a comprehensive understanding of the 3D object's form.

Here are the rough sketches of the top view and front view of a cylinder lying on its circular base:

Front View:

When looking at a cylinder lying on its circular base from the front, you see a rectangle. The width of the rectangle is the diameter of the base, and the height is the height of the cylinder.

          +-----------+
          |           |
          |           |  <-- Height
          |           |
          +-----------+
           <-- Diameter -->
        

Top View:

When looking down at a cylinder lying on its circular base from the top, you see the circular base.

              _____
             /     \
            (       )  <-- Circular Base
             \_____/
        

Here are the rough sketches of the side view and top view of a cone resting on its base:

Side View:

When looking at a cone resting on its base from the side, you see a triangle.

               /\    <-- Apex
              /  \
             /    \  <-- Slant Height
            /      \
           /________\  <-- Diameter of Base
        

Top View:

When looking down at a cone resting on its base from directly above, you see the circular base with the apex at the center. This can be represented as a circle with a dot in the middle, or lines drawn from the center to the edge to show the slant height/radius.

              _____
             / .   \
            (   .   )  <-- Circular Base with Apex at Center
             \_____/
        

A cross-section of a solid is the 2-dimensional (2D) shape formed when a 3-dimensional (3D) solid is intersected by a plane. It is essentially the shape you see when you slice through a solid object.

When you slice a carrot (assuming it's a cylinder) horizontally, the slicing plane is parallel to the circular bases of the cylinder.

The kind of cross-section you get by slicing a carrot (cylinder) horizontally is a circle.

A cross-section is the shape formed by the intersection of a plane with a solid object.

When a cube is cut by a plane that passes diagonally through two opposite vertices, the resulting cross-section is a rectangle.

This is because the plane containing the space diagonal connecting the two opposite vertices can also be chosen to contain two edges of the cube that are parallel to each other and skew to the space diagonal. The intersection of this plane with the faces of the cube forms a four-sided polygon (a quadrilateral). The sides of this quadrilateral are formed by the space diagonal itself and the lines where the plane intersects the faces. In the case where the plane contains two opposite edges, these intersection lines are the edges themselves and two face diagonals on opposite faces. This specific quadrilateral has opposite sides equal and parallel, and all angles are $90^\circ$, making it a rectangle.

For example, if we consider a cube with vertices at (0,0,0), (1,0,0), (0,1,0), (0,0,1), (1,1,0), (1,0,1), (0,1,1), (1,1,1), a plane cutting through opposite vertices (0,0,0) and (1,1,1) can pass through the vertices (1,1,0) and (0,0,1). The cross-section formed by the plane containing these four vertices is a rectangle with vertices (0,0,0), (1,1,0), (1,1,1), and (0,0,1).

Both oblique and isometric sketches are methods used to represent a 3-dimensional solid shape on a 2-dimensional surface (like paper or a screen). They aim to give a visual sense of depth and perspective, but they achieve this in different ways.

Oblique Sketch:

In an oblique sketch, one face of the object is drawn as its true shape and size, usually parallel to the drawing plane (the paper). The other faces are projected backwards at an angle, typically $30^\circ$ or $45^\circ$, to the first face. Edges parallel to the drawing plane are drawn at their true length, while receding edges (lines going into the page) can be drawn at their true length (cabinet projection) or half their length (cavalier projection) to enhance the visual effect of depth. Oblique sketches often appear distorted because the receding edges are not foreshortened realistically.

Isometric Sketch:

In an isometric sketch, the object is oriented so that three axes (representing length, width, and height) appear equally foreshortened and the angles between them are $120^\circ$. This results in three faces of the object being visible simultaneously, none of which are parallel to the drawing plane. All lines that are parallel to the main axes are drawn at their true length (isometric lines). Lines not parallel to the axes are called non-isometric lines and are drawn by locating their endpoints. Isometric sketches provide a more realistic representation of the object's dimensions and proportions compared to oblique sketches.

Key Differences:

1. Base Face: In oblique sketches, one face is drawn true-to-size and parallel to the plane. In isometric sketches, no face is parallel to the plane, and all visible faces are seen at an angle.

2. Angles: In oblique sketches, receding lines are drawn at an angle (commonly $30^\circ$ or $45^\circ$) to the front face. In isometric sketches, the axes are at $120^\circ$ to each other.

3. Lengths: In oblique sketches, receding lines can be drawn at full or half scale (can cause distortion). In isometric sketches, lines parallel to the axes (isometric lines) are drawn at true length, giving a consistent scale.

4. Realism: Isometric sketches generally provide a more realistic and less distorted view of the object's overall proportions than oblique sketches.

A real-life example of a solid shape that is a combination of a cylinder and a hemisphere is the positive terminal of a battery (like a common AA or AAA battery).

In this shape, the main body of the terminal is a cylinder, and the rounded tip on top is a hemisphere. The hemisphere is joined to one end of the cylinder.

Another example is a chemical storage tank with a rounded top (domed top), where the main tank is cylindrical and the top is a hemisphere.

Does a sphere have any flat faces?

No, a sphere does not have any flat faces. A sphere is a perfectly round 3D object where every point on its surface is equidistant from its center. Its entire surface is a single continuous curved surface.

Does a cylinder have any vertices?

No, a cylinder does not have any vertices. A vertex is a point where three or more edges meet. A cylinder has two flat circular faces and one curved lateral surface. It has edges where the flat bases meet the curved surface, but these edges are circles, and they do not meet at any points (vertices). Therefore, a cylinder has no vertices.

A pentagonal prism has 10 vertices.

A prism is a solid with two identical and parallel bases and rectangular faces connecting corresponding sides of the bases.

A pentagonal prism has a pentagon as its base. A pentagon is a polygon with 5 vertices.

Since there are two pentagonal bases in a pentagonal prism, the total number of vertices is the number of vertices on one base multiplied by 2.

The shape of the bases of a pentagonal prism is a pentagon.

An octagonal pyramid has an octagon as its base and triangular faces that meet at an apex.

An octagon is a polygon with 8 sides (and 8 vertices).

The edges of an octagonal pyramid consist of the edges on the base and the edges connecting the base vertices to the apex.

Number of edges on the base = 8

Number of edges connecting base vertices to apex = 8 (one from each vertex)

The lateral faces of any pyramid are always triangles. For an octagonal pyramid, there are 8 triangular lateral faces.

So, the shape of its lateral faces is a triangle.

The solid shape that has one vertex and one curved face is a Cone.

A cone has:

• One vertex (the apex)
• One curved lateral surface
• One flat circular base

While it also has a flat base, the description "one vertex and one curved face" uniquely identifies a cone among common solid shapes.

Recall that a polyhedron is a 3D solid bounded by flat surfaces that are polygons.

Is every prism a polyhedron?

Yes, every prism is a polyhedron.

Justification: A prism is defined by two parallel and congruent polygonal bases and lateral faces that are parallelograms (rectangles in the case of a right prism). Since both polygons (the bases) and parallelograms (the lateral faces) are flat surfaces, and all faces of a prism are polygons, a prism fits the definition of a polyhedron.

Is every pyramid a polyhedron?

Yes, every pyramid is a polyhedron.

Justification: A pyramid is defined by a polygonal base and triangular lateral faces that meet at a common vertex (apex). Since both polygons (the base) and triangles (the lateral faces) are flat surfaces, and all faces of a pyramid are polygons, a pyramid fits the definition of a polyhedron.

First, let's determine the number of faces for each solid shape:

Triangular Prism:

A triangular prism has two triangular bases and three rectangular lateral faces.

Number of faces = Number of triangular bases + Number of rectangular faces

Number of faces = 2 + 3

Number of faces = 5

Rectangular Pyramid:

A rectangular pyramid has one rectangular base and four triangular lateral faces that meet at an apex.

Number of faces = Number of rectangular base + Number of triangular faces

Number of faces = 1 + 4

Number of faces = 5

The difference between the number of faces of a triangular prism and a rectangular pyramid is:

Therefore, in terms of their number of faces, there is no difference; both a triangular prism and a rectangular pyramid have 5 faces.

When looking at the top of a solid and seeing a square, the solid must have a base or a top surface that is a square, and the rest of the structure must not extend beyond the boundaries of that square when viewed from directly above.

Possible shapes of the solid include:

1. A Cube: The top face of a cube is a square.

2. A Square Prism: The top base of a square prism is a square.

3. A Square Pyramid: The base of a square pyramid is a square, and the apex is directly above the center of the base. When viewed from the top, the outline seen is the square base.

4. A Rectangular Prism with a square base: If the base is a square, the top view will also be a square.

In general, any solid whose projection onto a horizontal plane is a square could have a square top view. The most common basic examples are a cube, square prism, or square pyramid.

A cross-section is the shape formed by the intersection of a plane with a solid object.

When you cut a cylinder with a plane that is parallel to its base, the resulting cross-section is a circle.

This is because the plane intersects all the lines that run parallel to the axis of the cylinder (the lines that connect the two bases) at the same 'level' relative to the bases. Since the bases are circles, any slice parallel to them will also produce a circular shape of the same size as the base.

The solid shape that has a circular base and tapers to a point at the top is called a Cone.

A cone has a single circular base and its surface tapers smoothly from the circumference of the base to a single point called the apex or vertex.

A cuboid is a 3D solid with 6 rectangular faces.

In a cuboid, opposite faces are identical (congruent). Since there are three pairs of opposite faces (top/bottom, front/back, left side/right side), a cuboid has 3 pairs of identical faces.

Each pair consists of two identical rectangular faces.

Differentiation between Prisms and Pyramids:

Both prisms and pyramids are types of polyhedra, which are 3D solids with flat polygonal faces. However, they differ in their structure, particularly concerning their bases, lateral faces, and vertices.

Prisms:

1. Bases: A prism has two bases. These bases are identical (congruent) polygons and are parallel to each other.

2. Lateral Faces: The faces connecting the two bases are called lateral faces. In a straight prism, these are rectangles. The number of lateral faces is equal to the number of sides of the polygonal base.

3. Vertices: Vertices exist at the corners of both bases. The total number of vertices is $2 \times$ (number of vertices in the base polygon).

Pyramids:

1. Bases: A pyramid has only one base. The base is a polygon.

2. Lateral Faces: The faces connecting the base to a single point (called the apex or vertex) are called lateral faces. These faces are always triangles. The number of lateral faces is equal to the number of sides of the base polygon.

3. Vertices: Vertices exist at the corners of the base polygon, and there is one additional vertex at the apex. The total number of vertices is (number of vertices in the base polygon) + 1.

Examples:

Example 1: Triangular Prism

A triangular prism has a triangle as its base.

• Faces: It has 2 triangular bases and 3 rectangular lateral faces. Total Faces = $2 + 3 = 5$.
• Edges: It has 3 edges on the bottom base, 3 edges on the top base, and 3 connecting edges. Total Edges = $3 + 3 + 3 = 9$.
• Vertices: It has 3 vertices on the bottom base and 3 vertices on the top base. Total Vertices = $3 + 3 = 6$.

Example 2: Square Pyramid

A square pyramid has a square as its base.

• Faces: It has 1 square base and 4 triangular lateral faces. Total Faces = $1 + 4 = 5$.
• Edges: It has 4 edges on the base and 4 edges connecting the base vertices to the apex. Total Edges = $4 + 4 = 8$.
• Vertices: It has 4 vertices on the base and 1 apex vertex. Total Vertices = $4 + 1 = 5$.

Explanation of Euler's Formula:

Euler's formula for polyhedrons is a mathematical relationship between the number of Faces (F), the number of Vertices (V), and the number of Edges (E) of any simple (convex or topologically equivalent to a sphere) polyhedron. The formula is given by:

This formula states that for any simple polyhedron, the sum of the number of its faces and vertices, minus the number of its edges, is always equal to 2.

Verification for a Cuboid:

A cuboid is a polyhedron with rectangular faces.

Number of Faces (F) = 6 (front, back, top, bottom, left, right)

Number of Vertices (V) = 8 (4 on the top face, 4 on the bottom face)

Number of Edges (E) = 12 (4 on the top face, 4 on the bottom face, 4 vertical connecting edges)

Substitute these values into Euler's formula $F + V - E = 2$:

Since $F + V - E = 2$, Euler's formula is verified for a cuboid.

Verification for a Triangular Pyramid:

A triangular pyramid (also known as a tetrahedron) has a triangular base and three triangular lateral faces.

Number of Faces (F) = 4 (1 triangular base + 3 triangular lateral faces)

Number of Vertices (V) = 4 (3 on the base + 1 apex)

Number of Edges (E) = 6 (3 on the base + 3 connecting base vertices to the apex)

Substitute these values into Euler's formula $F + V - E = 2$:

Since $F + V - E = 2$, Euler's formula is verified for a triangular pyramid.

Concept of 'Nets' for Solid Shapes:

A net of a 3-dimensional (3D) shape is a 2-dimensional (2D) pattern that can be folded along its edges to form the complete 3D shape. It is essentially the unfolded surface of the solid laid out flat in a single plane. Studying nets helps visualize the different faces of a solid and how they connect to each other.

Net for a Triangular Prism:

A triangular prism has two identical triangular bases and three rectangular lateral faces. A common net consists of the three rectangular faces laid out in a row, with one triangular base attached to the top edge of the central rectangle and the other triangular base attached to the bottom edge of that same central rectangle (or an adjacent rectangle). Other configurations are also possible.

Here is a rough sketch of a net for a triangular prism:

             /\   <-- Base 1 (Triangle)
            /  \
           /____\
          |      |
          |      |
          |      |   <-- Lateral Face 1 (Rectangle)
          |      |
          +------+------+------+
          |      |      |      |
          |      |      |      |
          |      |      |      |   <-- Lateral Faces 2 & 3 (Rectangles)
          |      |      |      |
          +------+------+------+
           \____/
            \  /
             \/   <-- Base 2 (Triangle)
        

Folding the Net to Form a Triangular Prism:

To fold this net into a triangular prism:

1. The three rectangular faces are the lateral faces of the prism. Imagine folding the outer two rectangles upwards along the vertical edges of the central rectangle. These three rectangles will form the sides of the prism.

2. The two triangular shapes are the bases of the prism. Fold these triangles upwards along the horizontal edges of the rectangular lateral faces to which they are attached.

3. As you fold the triangles, their edges will meet the corresponding edges of the rectangular lateral faces and also meet each other at the vertices, closing the shape and forming the top and bottom triangular bases of the prism.

The three rectangles will form the prism's sides, and the two triangles will form its parallel and congruent top and bottom ends (bases).

The solid formed by stacking three identical cubes one on top of the other is essentially a rectangular prism with a height three times its width and depth.

Here are rough sketches of its front view, side view, and top view:

Front View:

When viewed from the front, you see the stack of three cubes as a single rectangle that is three units high and one unit wide (where a unit is the side length of one cube).

          +-----+
          |     |  <-- Top cube face
          +-----+
          |     |  <-- Middle cube face
          +-----+
          |     |  <-- Bottom cube face
          +-----+
        

This view represents the solid as seen from directly in front, showing its height and width.

Side View:

Assuming the cubes are stacked vertically straight, the side view will be identical to the front view. You again see a rectangle that is three units high and one unit wide.

          +-----+
          |     |
          +-----+
          |     |
          +-----+
          |     |
          +-----+
        

This view represents the solid as seen from directly to its side, showing its height and depth (which appears as width in the 2D sketch).

Top View:

When viewed from directly above, you only see the top face of the uppermost cube. This is a square with sides equal to the side length of one cube (one unit by one unit).

          +-----+
          |     |
          +-----+
        

This view represents the solid as seen from directly above, showing its width and depth.

A cross-section of a solid is the 2D shape formed by the intersection of a plane with the solid. Different types of cuts (planes) through a cylinder will yield different cross-sectional shapes.

1. Horizontal Cut (Parallel to the Base):

When a cylinder is cut by a plane parallel to its circular base, the cross-section is a circle.

Description: The cutting plane is flat and runs horizontally across the cylinder, maintaining a constant distance from the bases.

Rough Sketch of Cut:

              _____
             /     \
            (       )
             \_____/   <-- Top Base
            +-------+
            |       |
            +-------+ <-- Horizontal Cut Plane
            |       |
            +-------+
             \_____/   <-- Bottom Base
        

Rough Sketch of Cross-section Shape:

              _____
             /     \
            (       )   <-- Resulting Shape
             \_____/
        

Shape of Cross-section: A Circle (identical in size to the bases).

2. Vertical Cut Through the Axis:

When a cylinder is cut by a plane that passes vertically through the central axis of the cylinder (perpendicular to the base), the cross-section is a rectangle.

Description: The cutting plane is flat and passes straight up and down through the exact center of the cylinder, from one base to the other.

Rough Sketch of Cut:

            +-------+
            |       |
            |   |   | <-- Vertical Cut Plane containing Axis
            |   |   |
            |   |   |
            +-------+
        

Rough Sketch of Cross-section Shape:

            +-----+
            |     |
            |     | <-- Resulting Shape
            |     |
            +-----+
        

Shape of Cross-section: A Rectangle (its width is the diameter of the base, and its height is the height of the cylinder).

3. Vertical Cut Not Through the Axis:

When a cylinder is cut by a plane that passes vertically but does not go through the central axis (still perpendicular to the base), the cross-section is also a rectangle, but typically a narrower one.

Description: The cutting plane is flat and passes straight up and down through the cylinder, but is offset from the center.

Rough Sketch of Cut:

            +-------+
            |       |
            |  |    | <-- Vertical Cut Plane (Off-center)
            |  |    |
            |  |    |
            +-------+
        

Rough Sketch of Cross-section Shape:

            +---+
            |   |
            |   | <-- Resulting Shape (A narrower rectangle)
            |   |
            +---+
        

Shape of Cross-section: A Rectangle (its width is a chord of the base circle, and its height is the height of the cylinder).

Other cross-sections are possible with angled cuts (e.g., an ellipse or a part of an ellipse if the cut is angled but doesn't pass through the base).

A cross-section of a solid is the 2D shape formed by the intersection of a plane with the solid. Different types of cuts (planes) through a cone will yield different cross-sectional shapes, which are also known as conic sections.

1. Horizontal Cut (Parallel to the Base):

When a cone is cut by a plane parallel to its circular base, the cross-section is a circle.

Description: The cutting plane is flat and runs horizontally across the cone, parallel to the base.

Rough Sketch of Cross-section Shape:

              _____
             /     \
            (       )   <-- Resulting Shape
             \_____/
        

Shape of Cross-section: A Circle.

2. Vertical Cut Through the Vertex:

When a cone is cut by a plane that passes vertically through the vertex (apex) and perpendicular to the base, the cross-section is a triangle.

Description: The cutting plane is flat and passes straight down through the pointed top (vertex) and includes the central axis of the cone.

Rough Sketch of Cross-section Shape:

               /\
              /  \
             /    \  <-- Resulting Shape
            /      \
           /________\
        

Shape of Cross-section: A Triangle (specifically, an isosceles triangle if it passes through the axis of a right cone).

3. Slant Cut Not Through the Vertex:

When a cone is cut by a plane that slants through the cone, is not parallel to the base, and does not pass through the vertex, the cross-section is an ellipse.

Description: The cutting plane is flat and angled. It slices through the cone intersecting all the generators (lines from the vertex to the base) but doesn't go through the vertex itself and is not parallel to the base.

Rough Sketch of Cross-section Shape:

               ____
             /      \
            /        \
           /          \  <-- Resulting Shape
           \          /
            \        /
             \______/
        

Shape of Cross-section: An Ellipse.

Other conic sections like a parabola or a hyperbola can also be obtained with specific angled cuts.

Both convex and concave polyhedrons are types of polyhedra, meaning they are 3D solids with flat polygonal faces. The difference lies in their internal structure and the shape formed by connecting any two points within or on their surface.

Convex Polyhedron:

A polyhedron is convex if, for any two points selected on the surface or within the polyhedron, the straight line segment connecting these two points lies entirely within or on the boundary of the polyhedron. In simpler terms, a convex polyhedron "bulges outwards" everywhere; it has no indentations.

All standard regular polyhedra (like cubes, tetrahedrons, octahedrons, dodecahedrons, icosahedrons) and most common everyday polyhedral shapes (like prisms and pyramids) are convex.

Concave Polyhedron:

A polyhedron is concave if there exists at least one pair of points on the surface or within the polyhedron such that the straight line segment connecting these two points passes outside the boundary of the polyhedron. Concave polyhedrons have at least one "indentation" or "cave" on their surface. At least one internal angle of a face, or the angle between two adjacent faces (dihedral angle), must be greater than $180^\circ$ (when viewed from inside the solid).

Real-Life Examples:

Example of a Convex Polyhedron: A standard dice (a cube).

Example of a Concave Polyhedron: A shape like a 'star' made from joining pyramids to the faces of a central solid, where the points of the star create indentations. Or, imagine two cubes joined at just an edge or a vertex - the combined shape is not a single polyhedron, but if you consider a single solid with an inward dent, like a 'L' shaped block made from cubes, it can be concave.

Visual Distinction:

The easiest way to visually distinguish between a convex and a concave polyhedron is to look for indentations or "dents" in the shape. A convex polyhedron will always appear smooth and rounded on its exterior corners and edges (even though the faces are flat polygons), and you cannot find any part of its surface that is "caved in". A concave polyhedron will have at least one inward curve or dent, making it possible to "hide" part of the solid from view if looking at it from certain angles.

Another visual test is to imagine shining a light from any direction; for a convex polyhedron, the entire surface that is facing the light source will be illuminated. For a concave polyhedron, some parts facing the light source might still be in shadow due to the indentations.

Determining Faces, Edges, and Vertices for the 'T' Shaped Solid:

To determine the number of Faces (F), Edges (E), and Vertices (V) for a solid shape formed by joining two (or more) polyhedrons, we can use the following approach:

1. Start by considering the total number of faces, edges, and vertices if the two rectangular prisms were separate.

2. Identify the surfaces (faces), lines (edges), and points (vertices) that are shared or become internal when the two prisms are joined. These shared elements are no longer part of the external count of faces, edges, or vertices of the combined solid.

3. Subtract the count of these shared/internal elements from the total initial count (sum of counts from separate prisms).

Let's assume the 'T' shape is formed by a vertical rectangular prism joined to the center of a horizontal rectangular prism. When the two prisms are joined, the face of one prism that connects to the face of the other becomes an internal surface and is no longer an external face of the combined solid. Similarly, the edges and vertices along the boundary of the joining faces become internal or merge.

Specifically, if the join is along a complete face of one prism that attaches to a complete face of the other of the same size:

• The two joined faces become internal. Subtract 2 from the total face count.
• The edges along the boundary of the joined faces merge and become internal edges. Subtract the number of edges on one of the joining faces from the total edge count.
• The vertices at the corners of the joined faces merge and become internal vertices. Subtract the number of vertices on one of the joining faces from the total vertex count.

If the join is partial (as in the case of a 'T' where the vertical bar sits on only a part of the horizontal bar's face), counting the external faces, edges, and vertices directly after visualizing the combined shape can be more reliable:

• Faces: Count all distinct, flat, external surfaces of the combined shape. A face of an original prism might become partially or fully internal.
• Edges: Count all the lines where two external faces meet. Be careful of original edges that become internal and edges formed on partially covered faces.
• Vertices: Count all the points where three or more external edges meet. Be careful of original vertices that become internal or merge.

For a typical 'T' made from joining two rectangular prisms, if done face-to-face without holes, the resulting solid is still a polyhedron.

As an example, consider a 'T' made by a vertical prism ($1 \times 1 \times 3$ units) and a horizontal prism ($3 \times 1 \times 1$ units), where the $1 \times 1$ top face of the vertical prism is centered on the $3 \times 1$ bottom face of the horizontal prism.

By carefully counting the external elements:

Number of Faces (F) = 10

Number of Vertices (V) = 12

Number of Edges (E) = 20

Would Euler's formula apply to this solid?

Yes, Euler's formula would apply to this solid.

Explanation:

Euler's formula ($F + V - E = 2$) applies to simple polyhedrons. A simple polyhedron is one that is topologically equivalent to a sphere; it is a single, connected solid with no holes (like tunnels). The solid shape formed by joining two rectangular prisms face-to-face, even if the join is partial (like the 'T' shape described), results in a shape whose entire surface is a single continuous boundary without holes.

Since the resulting 'T' shaped solid is a polyhedron (bounded by flat polygonal surfaces) and is a simple, connected solid without holes, Euler's formula $F + V - E = 2$ holds true for it.

Verification using the example counts (F=10, V=12, E=20):

Since $F + V - E = 2$, Euler's formula is verified for this composite solid.

Concept of a Net:

A net of a 3-dimensional (3D) shape is a 2-dimensional (2D) pattern that, when folded along its edges, forms the 3D shape. It shows all the faces of the solid laid out flat.

Rough Sketch of the Net of a Square Pyramid:

A square pyramid has one square base and four triangular lateral faces. A common net consists of the square base with the four triangular faces attached to each of its sides.

            /\
           /  \
          /    \
         /______\   <-- Lateral Face (Triangle)
         |      |
         |      |   <-- Base (Square)
         |      |
         +------+------+------+
         |      |      |      |  <-- Lateral Faces (Triangles)
         |      |      |      |
         +------+------+------+
                |      |
                |      |
                |______|   <-- Lateral Face (Triangle)
        

(Note: The four triangles should ideally be congruent isosceles triangles for a right square pyramid, with the base of each triangle matching the side length of the square).

Folding the Net to Form the Pyramid:

To fold this net into a square pyramid:

1. The central square shape is the base of the pyramid.

2. The four triangular shapes attached to the sides of the square are the lateral faces. Imagine lifting these four triangles upwards.

3. Fold each triangular face along the edge where it is attached to the square base. As you fold them up, the unattached edges of the adjacent triangular faces will meet.

4. Join these meeting edges. The vertices opposite the base of the four triangles will meet at a single point above the center of the square base. This point is the apex of the pyramid.

The four triangles form the sloping sides, and the square forms the bottom base.

Shapes of the faces in the net:

The shapes of the faces in the net of a square pyramid are:

1. One Square (the base)

2. Four Triangles (the lateral faces)

Characteristics of a Sphere:

A sphere is a perfectly round 3-dimensional solid shape. Its defining characteristic is that every point on its surface is the same distance from a central point called the center. It has no flat faces, no straight edges, and no vertices. Its entire boundary is a single, continuous curved surface.

Differences from other common solid shapes:

• Sphere vs. Cube: A cube has 6 flat square faces, 12 straight edges, and 8 vertices. A sphere has none of these; it is entirely curved.
• Sphere vs. Cylinder: A cylinder has two flat circular bases, one curved lateral surface, and two circular edges. A sphere has no flat faces, no edges, and no vertices.
• Sphere vs. Cone: A cone has one flat circular base, one curved lateral surface, one circular edge, and one vertex (apex). A sphere has no flat faces, no edges, and no vertices.

The fundamental difference is that spheres are bounded solely by a curved surface, whereas cubes, cylinders, and cones are bounded by combinations of flat (polygonal or circular) faces and curved surfaces, containing straight edges and/or vertices.

Can you apply Euler's formula to a sphere?

No, you cannot directly apply Euler's formula ($F + V - E = 2$) to a sphere in the same way you apply it to polyhedrons.

Explanation:

Euler's formula ($F + V - E = 2$) is specifically defined for polyhedrons, which are 3D solids bounded by flat polygonal faces. The terms F, V, and E in the formula represent the number of flat Faces, the number of Vertices (points where edges meet), and the number of straight Edges (lines where faces meet) of a polyhedron.

A sphere has no flat faces (F=0), no straight edges (E=0), and no vertices (V=0) according to these definitions for polyhedrons. If we were to substitute these values into the formula, we would get $0 + 0 - 0 = 0$, which does not equal 2.

The surface of a sphere is topologically equivalent to the surface of any convex polyhedron (both are like a stretched sphere with no holes). In more advanced topology, Euler's characteristic ($\chi$) is a concept that generalizes $F+V-E$. For any polyhedron topologically equivalent to a sphere, $\chi = F+V-E = 2$. For a sphere, if one were to triangulate its surface (divide it into arbitrary "faces", "edges", and "vertices"), the relationship $F+V-E$ would indeed equal 2, demonstrating its underlying topological property. However, in the context of elementary geometry, where F, V, and E refer to the inherent flat faces, straight edges, and distinct vertices of the solid's natural structure, a sphere does not possess these features, and the formula $F+V-E=2$ as applied to counting these elements does not hold.

Understanding Different Views of a 3D Shape:

Visualizing a 3-dimensional (3D) shape solely from a single 2-dimensional (2D) representation can be challenging. Different 'views' of a 3D shape, such as the front view, side view, and top view, are 2D projections of the solid as seen from specific directions.

Understanding these different views helps in visualizing the 3D shape better because each view provides unique information about the shape's dimensions and features from a particular perspective. The front view typically shows height and width, the side view shows height and depth, and the top view shows width and depth. By combining the information from these standard orthogonal views, one can mentally construct a complete picture of the solid's form, proportions, and details, including features that might be hidden in a single perspective view.

They help clarify:

• The exact dimensions (length, width, height).
• The relative positions of features (holes, extrusions, etc.).
• The overall structure and complexity of the object.

Real-Life Scenario:

A crucial real-life scenario where looking at different views of a 3D shape is important is in Engineering and Architecture.

Engineers and architects use technical drawings (blueprints or plans) to communicate designs for buildings, machinery parts, products, etc. These drawings almost always include multiple views (orthographic projections) – typically front, top, and side views. For example:

• In Architecture, floor plans (top view) show the layout of rooms, walls, doors, and windows from above. Elevations (front and side views) show the external appearance of the building, including heights, rooflines, and facade details. Sections (cuts through the building) show internal structure and heights. Without these different views, it would be impossible for contractors to build the structure accurately or for clients to fully understand the design.
• In Mechanical Engineering, technical drawings of machine parts show multiple views to specify precise dimensions, tolerances, and features like holes, threads, and curves from all necessary angles. A single view would not contain enough information for manufacturing.

In these fields, the ability to read and interpret multiple views is fundamental for design, manufacturing, construction, and assembly.

The solid is formed by placing the circular base of the cone directly onto the top circular base of the cylinder, as they have the same radius. The two faces where they join become internal to the combined solid and are no longer considered external faces.

Let's describe the external faces, edges, and vertices of this combined solid:

Faces:

• The bottom base of the cylinder is an external flat face (a circle).
• The lateral surface of the cylinder is an external curved face.
• The lateral surface of the cone is an external curved face.
• The base of the cone and the top base of the cylinder are internal and not counted as external faces.

Total external Faces = 3.

Edges:

• The bottom circular edge of the cylinder is an external edge.
• The circular edge where the cone base meets the cylinder top becomes an internal boundary and is not counted as an external edge.
• A standard cone has no straight lateral edges; its lateral surface is smooth.

Total external Edges = 1 (the bottom circular edge).

Vertices:

• The cylinder has no vertices.
• The apex of the cone is the only vertex of the combined solid.

Total external Vertices = 1 (the apex of the cone).

Summary of Faces:

• Number of Curved Faces = 2 (cylinder lateral surface + cone lateral surface).
• Number of Flat Faces = 1 (bottom base of the cylinder).