| Classwise Concept with Examples | ||||||
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| 6th | 7th | 8th | 9th | 10th | 11th | 12th |
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| 2-Dimensional and 3-Dimensional Shapes | Views of 3-D Shapes | Polyhedron and Terms Related to it |
Chapter 10 Visualising Solid Shapes (Concepts)
Welcome back to the intriguing world of three dimensions! This chapter, Visualising Solid Shapes, significantly refines and extends the foundational concepts introduced in Class 7, aiming to sharpen our ability to perceive, interpret, and represent three-dimensional (3D) objects within the constraints of a two-dimensional (2D) plane, like this page. We will revisit familiar ideas but with greater depth and formality, enhancing our spatial intuition and geometric understanding.
We begin by reinforcing our understanding of the essential components that constitute solid shapes. Recall these fundamental elements:
- Faces: These are the flat (or sometimes curved, as in cylinders or cones) surfaces that enclose the solid. Think of them as the 'skin' of the shape.
- Edges: These are the line segments formed where two faces meet. They represent the 'seams' or 'folds' of the solid structure.
- Vertices (singular: Vertex): These are the sharp corners of the solid, representing the points where three or more edges intersect.
We will re-examine common 3D shapes like cubes, cuboids, cylinders, cones, spheres, and various pyramids, practicing the identification of their respective faces, edges, and vertices. This leads us to a more formal definition of Polyhedrons (or polyhedra) – solid shapes constructed entirely from flat polygonal faces, straight edges, and sharp vertices. Cubes, cuboids, prisms, and pyramids are prime examples of polyhedrons.
A remarkable and central relationship governing polyhedrons is introduced: Euler's Formula (also known as Euler's Polyhedron Formula). This elegant formula provides a constant relationship between the number of faces (F), vertices (V), and edges (E) for any simple, convex polyhedron: $$ F + V - E = 2 $$ We will explore this formula and practice verifying it for various polyhedrons. For example, a cube has $F=6$, $V=8$, and $E=12$. Plugging these into the formula gives $6 + 8 - 12 = 14 - 12 = 2$, confirming Euler's relationship. This formula reveals a fundamental topological property invariant across a wide range of solid shapes.
The challenge of representing these 3D objects on a 2D surface is further explored. We revisit the techniques of drawing oblique sketches (easy but often dimensionally distorted) and isometric sketches (using isometric dot paper for better proportional representation and perspective). The goal now is to improve the quality and accuracy of these representations. A significant focus is placed on understanding and drawing different standard views of 3D objects, known as orthographic projections. These typically include:
- The Top View (as seen from directly above)
- The Front View (as seen from directly in front)
- The Side View (as seen from the side, usually left or right)
We will practice generating these 2D views from given 3D objects (like arrangements of stacked cubes) and, conversely, developing the skill to visualize or even reconstruct the original 3D object when provided with its distinct 2D views. The chapter may also touch upon the practical application of spatial visualization through map interpretation, understanding scale, and relating the positions of landmarks. Ultimately, this chapter aims to significantly boost your spatial reasoning skills – the ability to mentally manipulate 3D objects and to fluently translate between the 3D world and its various 2D representations, including nets, sketches, views, and the underlying structure revealed by formulas like Euler's.
2-Dimensional and 3-Dimensional Shapes
In geometry, we study shapes and their properties. We can classify these shapes based on the number of dimensions they occupy in space. You are already familiar with basic shapes like squares and circles, and also with solid objects like boxes and balls. This section will formally distinguish between 2-dimensional and 3-dimensional shapes.
2-Dimensional (2-D) Shapes or Plane Figures
2-Dimensional shapes, also called plane figures, are shapes that lie entirely within a single flat surface or a plane. They have only two fundamental dimensions: length and width (or breadth). They do not have any thickness or depth.
Imagine drawing a shape on a sheet of paper. The paper itself can be considered a plane, and the shapes you draw on it (like a square, a circle, etc.) are 2-dimensional. They have area but no volume.
Examples of 2-D shapes:
- Square: Has length and width (which are equal).
- Rectangle: Has length and width.
- Triangle: Has base and height (dimensions used to calculate area).
- Circle: Has a radius which defines its extent in two dimensions.
- Quadrilateral: Any four-sided polygon lying in a plane.
- Polygon: Any closed figure made of straight line segments lying in a plane (e.g., pentagon, hexagon, octagon).
You can fully represent a 2-D shape on a flat surface.
3-Dimensional (3-D) Shapes or Solid Shapes
3-Dimensional shapes, also known as solid shapes or solids, are shapes that do not lie entirely in a single plane. They occupy space. They have three fundamental dimensions: length, width (or breadth), and height (or depth or thickness).
Objects in the real world around us are 3-D shapes. They have volume.
Examples of 3-D shapes:
- Cube: Has length, width, and height (all equal edges). Example: a dice.
- Cuboid: Has length, width, and height (like a rectangular box). Example: a brick.
- Cylinder: Has a circular base and height. Example: a can.
- Cone: Has a circular base and tapers to an apex (height). Example: an ice cream cone.
- Sphere: A perfectly round solid. Example: a ball.
- Pyramid: Has a polygonal base and triangular faces meeting at an apex (height). Example: Egyptian pyramids.
- Prism: Has two identical parallel polygonal bases connected by rectangular faces (height). Example: a triangular prism looks like a Toblerone bar.
You cannot represent a 3-D shape on a 2-D surface perfectly without losing information or using techniques like perspective drawing or showing different views (which we will discuss later).
The surfaces of 3-D shapes are often made up of 2-D shapes. For example:
- A cuboid has 6 rectangular faces (rectangles are 2-D).
- A cube has 6 square faces (squares are 2-D).
- A cylinder has 2 circular bases (circles are 2-D) and one curved surface.
- A pyramid has a polygonal base (a 2-D polygon) and triangular faces (triangles are 2-D).
Connecting 2-D and 3-D Shapes (Nets)
A net is a 2-D shape that can be folded along lines to form a 3-D shape. It shows all the faces of a 3-D solid laid out flat in a single plane.
Example: The net of a cube consists of six squares connected in a way that they can be folded to form a cube.
Example: The net of a cylinder consists of two circles (the bases) and a rectangle (the curved surface unrolled).
Nets help us visualise the relationship between the 2-D faces and the 3-D solid they form.
Example 1. Classify the following objects as 2-D or 3-D shapes: A photograph, a book, a coin, the surface of a table, a ball.
Answer:
- A photograph: It has length and width but negligible thickness. It is considered a 2-D shape (a rectangle or square).
- A book: It has length, width, and significant thickness/height. It is a 3-D shape (a cuboid).
- A coin: While thin, it has a measurable thickness in addition to its circular face. It is a 3-D shape (a cylinder).
- The surface of a table: This refers only to the flat top part, ignoring the thickness. It is considered a 2-D shape (a rectangle or square).
- A ball: It occupies space and has three dimensions. It is a 3-D shape (a sphere).
Views of 3-D Shapes
Since 3-dimensional (3-D) shapes occupy space and have depth, we cannot represent them perfectly on a 2-dimensional (2-D) surface like a piece of paper or a screen without losing some information or using techniques like perspective. One common way to understand and represent 3-D shapes in 2-D is by drawing their different views.
A 3-D object looks different when viewed from different angles or positions. The view from a specific direction gives us a 2-D representation of the object as seen from that perspective.
Common Views of 3-D Shapes
The most standard views used to represent 3-D objects are obtained by looking at the object from directions that are perpendicular to its main faces or axes. The common views are:
- Front View: The view of the object as seen from the front. This typically shows the length and height of the object or its parts facing forward.
- Side View: The view of the object as seen from one of its sides (usually the right side view is shown if not specified, but left side view is also possible). This view typically shows the width (or depth) and height of the object or its parts facing the side.
- Top View (or Plan View): The view of the object as seen from directly above. This view typically shows the length and width (or depth) of the object or its parts as projected onto the horizontal plane.
Imagine placing a 3-D object inside a transparent box and looking at it from the front, side, and top. The shape you would see projected onto the sides of the box are its views.
Examples of Different Views of Solid Shapes
Let's look at the front, side, and top views of some common 3-D shapes.
Example 1. Consider a cuboid. Assume the longest edge is the length, the shorter horizontal edge is the width, and the vertical edge is the height. The front face is the rectangle defined by length and height. The side face is defined by width and height. The top face (base) is defined by length and width.
What are its front, side, and top views?
Answer:
Based on the assumption about front, side, and top faces:
Front View: When you look at a cuboid from the front, you see a rectangle. The dimensions of this rectangle are the length and the height of the cuboid.
Side View: When you look at the cuboid from the side (e.g., the right side), you see a rectangle. The dimensions of this rectangle are the width (or depth) and the height of the cuboid.
Top View: When you look at the cuboid from directly above, you see a rectangle. The dimensions of this rectangle are the length and the width (or depth) of the cuboid.
In all cases, the views of a cuboid are rectangles. For a cube (a special cuboid), all three views would be squares of the same size.
Example 2. Consider a cylinder resting on its circular base.
What are its front, side, and top views?
Answer:
Assume the cylinder is standing vertically on its base.
Front View: When you look at a vertical cylinder from the front, you see a rectangle. The width of this rectangle is the diameter of the circular base, and the height is the height of the cylinder.
Side View: Due to the cylindrical shape, looking from any side perpendicular to the base gives the same view as the front view – a rectangle with the same dimensions (diameter as width and cylinder height as height).
Top View: When you look at a vertical cylinder from directly above, you see its circular base.
Example 3. Draw the different views (Front, Side, and Top) of the object shown below. The object is made by stacking identical cubes.
Assume the front direction is as indicated by your perspective in the image (the side with two cubes at the bottom and one above the left one).
Answer:
Let's analyse the structure. There are two cubes side-by-side in the bottom layer, and one cube stacked on top of the left cube in the bottom layer.
Front View: Imagine standing in front of the structure. You would see the faces of the cubes that are visible from the front. In this case, you see two squares at the bottom (the front faces of the two bottom cubes) and one square stacked on top of the left bottom square (the front face of the top cube).
Side View: Imagine standing to the right of the structure. You would see the faces of the cubes visible from the right side. In this case, you see the right face of the right bottom cube, and the right face of the cube stacked above it. These two faces appear one above the other.
Top View: Imagine looking down on the structure from directly above. You would see the top faces of the cubes. You see the top face of the left bottom cube and the top face of the right bottom cube. The top face of the cube stacked on the left bottom one aligns with the top face of the left bottom cube from this perspective.
Drawing and interpreting different views of 3-D shapes is a fundamental skill in technical drawing, architecture, and understanding how objects are represented on flat surfaces.
Polyhedron and Terms Related to it
In our previous discussions about 3-dimensional shapes, we saw that many solid shapes are bounded by flat surfaces (like cubes, cuboids) or a combination of flat and curved surfaces (like cylinders, cones) or only curved surfaces (like spheres). This section focuses on 3-D shapes that are made up entirely of flat faces.
What is a Polyhedron?
A polyhedron (plural: polyhedra or polyhedrons) is a three-dimensional solid figure whose surface is composed of a finite number of flat, polygonal faces. These faces are joined along straight line segments called edges, and the edges meet at points called vertices.
In simpler terms, a polyhedron is a solid shape whose 'skin' is made entirely of polygons.
Examples of polyhedra:
- Cube (faces are squares)
- Cuboid (faces are rectangles)
- Triangular Prism (faces are triangles and rectangles)
- Square Pyramid (faces are a square and triangles)
- Triangular Pyramid (Tetrahedron - faces are triangles)
- Octahedron (faces are triangles)
Examples of 3-D shapes that are NOT polyhedra (because they have curved surfaces):
- Cylinder: Has circular bases and a curved lateral surface.
- Cone: Has a circular base and a curved lateral surface.
- Sphere: Has an entirely curved surface.
Terms Related to Polyhedra
As mentioned in the definition, polyhedra are made up of three main components:
- Faces (F): These are the flat surfaces that form the boundary of the polyhedron. Each face is a polygon (triangle, quadrilateral, pentagon, etc.).
- Edges (E): These are the straight line segments where two faces meet.
- Vertices (V): These are the points where three or more edges meet. These are the 'corners' of the polyhedron.
Counting Faces, Edges, and Vertices of Common Polyhedra
Let's count the number of faces, edges, and vertices for some common polyhedra:
| Polyhedron | Faces (F) | Edges (E) | Vertices (V) |
|---|---|---|---|
| Cube | 6 (all square) | 12 | 8 |
| Cuboid | 6 (all rectangular) | 12 | 8 |
| Triangular Prism | 5 (2 triangular bases, 3 rectangular lateral faces) | 9 (3 on each base, 3 connecting bases) | 6 (3 on each base) |
| Square Pyramid | 5 (1 square base, 4 triangular lateral faces) | 8 (4 on the base, 4 connecting base to apex) | 5 (4 on the base, 1 apex) |
| Triangular Pyramid (Tetrahedron) | 4 (all triangular) | 6 | 4 |
| Pentagonal Prism | 7 (2 pentagonal bases, 5 rectangular lateral faces) | 15 (5 on each base, 5 connecting bases) | 10 (5 on each base) |
| Hexagonal Pyramid | 7 (1 hexagonal base, 6 triangular lateral faces) | 12 (6 on the base, 6 connecting base to apex) | 7 (6 on the base, 1 apex) |
Convex and Concave Polyhedra
Polyhedra can be classified as convex or concave, similar to polygons.
- Convex Polyhedron: A polyhedron is convex if, for every face, the entire polyhedron lies on one side of the plane containing that face. Imagine placing the polyhedron on a flat surface that extends infinitely; if the entire solid remains on one side of the surface, it is convex. All standard prisms and pyramids are convex.
- Concave Polyhedron: A polyhedron is concave if it is not convex. This means there is at least one face such that the plane containing that face cuts through the polyhedron, leaving part of the solid on both sides of the plane. Concave polyhedra 'dent inwards'.
Regular Polyhedra
A convex polyhedron is called regular if two conditions are met:
- Its faces are made up of regular polygons of the same type.
- The same number of faces meet at each vertex.
There are only five types of regular convex polyhedra, known as the Platonic Solids:
- Tetrahedron: 4 faces (equilateral triangles), 3 faces meet at each vertex.
- Cube (Hexahedron): 6 faces (squares), 3 faces meet at each vertex.
- Octahedron: 8 faces (equilateral triangles), 4 faces meet at each vertex.
- Dodecahedron: 12 faces (regular pentagons), 3 faces meet at each vertex.
- Icosahedron: 20 faces (equilateral triangles), 5 faces meet at each vertex.
These are highly symmetrical shapes.
Prisms and Pyramids (Detailed Definition)
Let's revisit the definitions of prisms and pyramids more formally, as they are common types of polyhedra.
- Prism: A prism is a polyhedron formed by two congruent (identical in shape and size) polygonal bases that are parallel to each other, and whose other faces (lateral faces) are parallelograms (typically rectangles in a right prism) connecting the corresponding sides of the bases. Prisms are named after the shape of their bases (e.g., triangular prism, square prism, pentagonal prism). A rectangular prism is a cuboid.
- Pyramid: A pyramid is a polyhedron formed by a polygonal base and triangular faces (lateral faces) that meet at a common vertex called the apex. Pyramids are named after the shape of their bases (e.g., triangular pyramid, square pyramid, hexagonal pyramid). A triangular pyramid with all four faces being equilateral triangles is a regular tetrahedron.
Euler's Formula for Polyhedra
For any convex polyhedron, there is a remarkable relationship between the number of faces (F), the number of vertices (V), and the number of edges (E). This relationship is given by Euler's Formula (also known as Euler's Polyhedral Formula):
F + V $-$ E = 2
... (i)
This formula holds true for all convex polyhedra.
Let's verify this using the counts from the table above:
- Cube: F=6, V=8, E=12. F + V - E = $6 + 8 - 12 = 14 - 12 = 2$. Verified.
- Cuboid: F=6, V=8, E=12. F + V - E = $6 + 8 - 12 = 14 - 12 = 2$. Verified.
- Triangular Prism: F=5, V=6, E=9. F + V - E = $5 + 6 - 9 = 11 - 9 = 2$. Verified.
- Square Pyramid: F=5, V=5, E=8. F + V - E = $5 + 5 - 8 = 10 - 8 = 2$. Verified.
- Triangular Pyramid: F=4, V=4, E=6. F + V - E = $4 + 4 - 6 = 8 - 6 = 2$. Verified.
- Pentagonal Prism: F=7, V=10, E=15. F + V - E = $7 + 10 - 15 = 17 - 15 = 2$. Verified.
- Hexagonal Pyramid: F=7, V=7, E=12. F + V - E = $7 + 7 - 12 = 14 - 12 = 2$. Verified.
Euler's formula is a fundamental property of convex polyhedra and can be used to find the number of faces, vertices, or edges if the other two quantities are known.
Example 1. A polyhedron has 10 faces and 15 edges. How many vertices does it have?
Answer:
Given:
- Number of Faces (F) = 10.
- Number of Edges (E) = 15.
To Find:
- Number of Vertices (V).
Solution:
We can use Euler's Formula for convex polyhedra, F + V - E = 2.
F + V $-$ E = 2
[Euler's Formula]
Substitute the given values for F and E into the formula:
$10 + V - 15 = 2$
[Substituting given values]
Simplify the left side:
$V + (10 - 15) = 2$
$V - 5 = 2$
Solve for V by adding 5 to both sides (or transposing -5 to the RHS):
$V = 2 + 5$
[Adding 5 to both sides]
$V = 7$
The polyhedron has 7 vertices.