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| Different Solid Shapes and Terms Related to it | Drawing Solids on a Flat Surface | Visualising Different Sections of a Solid |
Chapter 15 Visualising Solid Shapes (Concepts)
This chapter embarks on a fascinating journey into the third dimension, aiming to significantly enhance your ability to visualize and represent three-dimensional (3D) objects, often referred to as solid shapes. While our previous studies have largely focused on plane figures confined to two dimensions (2D), like squares, circles, and triangles – shapes possessing only length and breadth – the world around us is overwhelmingly three-dimensional. This section bridges the gap, introducing tools and techniques to understand and depict objects that possess length, breadth, and crucially, height or depth, such as cubes, cuboids, cylinders, cones, and spheres.
To effectively analyze and describe these solid shapes, we first need to identify their fundamental components. The key elements are:
- Faces: These are the flat surfaces (or curved surfaces in shapes like cylinders, cones, and spheres) that form the 'skin' of the solid object. For example, a cube has 6 square faces.
- Edges: These are the straight line segments formed where two faces meet. Think of them as the 'seams' or 'folds' of the shape. A cube, for instance, has 12 edges.
- Vertices (singular: Vertex): These are the sharp points or corners of the solid shape, formed where three or more edges intersect. A cube has 8 vertices.
Understanding these components (Faces, Edges, Vertices - often abbreviated as F, E, V) is essential for describing and classifying solid shapes.
A crucial concept introduced for understanding the construction of 3D shapes from 2D representations is the idea of Nets. A net is essentially a flat, two-dimensional pattern or layout that can be cut out and folded along its edges to create a specific three-dimensional solid figure without any overlaps. Imagine 'unfolding' a cardboard box – the resulting flat shape is its net. Learning to recognize and draw nets for common solids like cubes, cuboids, cylinders, cones, and various pyramids is a key skill developed in this chapter. Working with nets provides invaluable insight into the surface area and structure of 3D objects.
Since we typically represent shapes on flat surfaces like paper, the chapter explores different methods for drawing or sketching 3D objects in 2D. Two primary techniques are commonly taught:
- Oblique Sketches: These are relatively simple to draw freehand. While they attempt to show the three-dimensional nature, they often don't maintain accurate lengths or angles, especially for lines representing depth. Parallel edges of the 3D object are typically drawn as parallel lines in the sketch, but the overall proportions might appear distorted.
- Isometric Sketches: These provide a more realistic and dimensionally proportional representation of a 3D object. They are usually drawn using special isometric dot paper, where dots are arranged in equilateral triangles. Isometric sketches maintain the relative measurements along the three principal axes, giving a much better visual sense of the object's actual shape and size.
Another vital aspect of spatial visualization involves understanding how a 3D object appears from different viewpoints. We learn to interpret and draw the standard orthographic projections: the Top view (looking down from directly above), the Front view (looking from directly in front), and the Side view (usually the view from the left or right). This skill is critical in fields like engineering and design. Additionally, the chapter may briefly introduce the concept of visualizing cross-sections – the 2D shape revealed when a solid object is imagined to be sliced through, either horizontally or vertically. For example, slicing a cylinder horizontally yields a circle, while a vertical slice produces a rectangle.
Finally, for a special class of solids known as polyhedra (solids composed entirely of flat polygonal faces, straight edges, and sharp vertices, like cubes, prisms, and pyramids), a remarkable mathematical relationship is often introduced: Euler's Formula. This formula elegantly connects the number of Faces (F), Vertices (V), and Edges (E) of any simple, convex polyhedron through the equation: $F + V - E = 2$. Verifying this formula for different polyhedra reinforces the understanding of their structural components. Overall, this chapter places strong emphasis on developing spatial reasoning, the ability to mentally picture and manipulate 3D objects, and interpret their 2D representations effectively.
Different Solid Shapes and Terms Related to it
In your earlier studies, you have explored two-dimensional (2D) shapes like squares, rectangles, triangles, and circles. These are flat shapes that exist entirely within a plane and are described by two dimensions: length and breadth (or width).
Now, we move into the world of three-dimensional (3D) shapes, also known as solid shapes. These shapes occupy space and are described by three dimensions: length, breadth (or width), and height (or depth). They cannot lie completely flat on a single plane.
Understanding Solid Shapes (3D Shapes)
Solid shapes are all around us in our everyday life. From the room you are sitting in to the objects on your desk, most things are solid shapes. They have 'thickness' or 'depth' in addition to length and width.
Visualising 3D shapes is important to understand their properties and how they differ from 2D shapes.
Terms Related to Solid Shapes
Solid shapes are made up of specific parts, which we describe using special terms: Faces, Edges, and Vertices. These terms are crucial for describing and classifying different solid shapes.
- Faces: A face is a single flat (planar) surface of a solid shape. In shapes like cubes and cuboids, the faces are polygons (like squares or rectangles). Some solid shapes, like cylinders and cones, have curved surfaces, which are sometimes also referred to as faces in a broader sense, though technically, the term 'face' in polyhedra refers only to flat polygonal surfaces. We will primarily focus on flat faces unless specified.
- Edges: An edge is a line segment where two faces of a solid shape meet. Think of the lines forming the skeleton of a box – these are the edges.
- Vertices: A vertex (plural: vertices) is a point where three or more edges meet. These are the 'corners' of the solid shape.
Let's look at a cuboid to clearly see these parts:
Here, the shaded rectangular areas are the faces, the straight lines connecting them are the edges, and the points where the edges meet are the vertices.
Types of Solid Shapes and Their Properties
Let's examine some common solid shapes and identify their faces, edges, and vertices. We will also note if they have flat or curved surfaces.
1. Cube:
A cube is a solid shape that is formed by six identical square faces. All its edges are of equal length.
- Number of Faces: 6 (all flat and square)
- Number of Edges: 12 (all straight)
- Number of Vertices: 8
Examples: A standard dice, a sugar cube, a small cardboard box of perfect square shape.
2. Cuboid:
A cuboid is a solid shape whose faces are rectangles. It has six rectangular faces, where opposite faces are identical. A cube is a special case of a cuboid where all faces are squares.
- Number of Faces: 6 (all flat and rectangular)
- Number of Edges: 12 (all straight)
- Number of Vertices: 8
Examples: A brick, a book, a matchbox, a room.
3. Cylinder:
A cylinder is a solid shape with two identical, flat circular bases and one curved surface connecting them. It is different from cubes and cuboids as it has curved surfaces and edges.
- Number of Flat Faces: 2 (circular bases)
- Number of Curved Surfaces: 1
- Number of Edges: 2 (curved edges where the circular bases meet the curved surface)
- Number of Vertices: 0
Examples: A cold drink can, a drum, a piece of pipe, a pillar.
4. Cone:
A cone is a solid shape with one flat circular base and one curved surface that tapers to a single point called the apex or vertex.
- Number of Flat Faces: 1 (circular base)
- Number of Curved Surfaces: 1
- Number of Edges: 1 (the curved edge of the circular base)
- Number of Vertices: 1 (the apex)
Examples: An ice cream cone, a birthday cap, a funnel.
5. Sphere:
A sphere is a perfectly round solid shape where every point on its surface is the same distance from its center. It has no flat surfaces, straight edges, or vertices.
- Number of Faces: 1 (the entire surface is one continuous curved surface)
- Number of Edges: 0
- Number of Vertices: 0
Examples: A ball, a marble, a globe.
6. Pyramid:
A pyramid is a solid shape with a polygon as its base and triangular faces that meet at a common vertex, called the apex. Pyramids are named based on the shape of their base.
Square Pyramid: A pyramid with a square base.
- Number of Faces: 5 (1 square base + 4 triangular faces)
- Number of Edges: 8 (4 base edges + 4 edges connecting base vertices to the apex)
- Number of Vertices: 5 (4 base vertices + 1 apex vertex)
Triangular Pyramid (Tetrahedron): A pyramid with a triangular base.
- Number of Faces: 4 (1 triangular base + 3 triangular faces)
- Number of Edges: 6 (3 in the base + 3 connecting base vertices to the apex)
- Number of Vertices: 4 (3 in the base + 1 at the apex)
7. Prism:
A prism is a solid shape with two identical and parallel polygonal bases. The faces connecting the corresponding sides of the bases are parallelograms (in a right prism, these are rectangles). Prisms are named based on the shape of their bases.
Triangular Prism: A prism with triangular bases.
- Number of Faces: 5 (2 triangular bases + 3 rectangular connecting faces)
- Number of Edges: 9 (3 edges in each base + 3 edges connecting the bases)
- Number of Vertices: 6 (3 vertices in each base)
Note: A rectangular prism is the same as a cuboid.
Summary of Properties of Common Solid Shapes
Here is a table summarising the key properties (Faces, Edges, and Vertices) for the solid shapes we discussed:
| Solid Shape | Number of Faces (Flat) | Number of Edges | Number of Vertices |
|---|---|---|---|
| Cube | 6 | 12 | 8 |
| Cuboid | 6 | 12 | 8 |
| Cylinder | 2 (flat) + 1 (curved) = 3 surfaces | 2 (curved) | 0 |
| Cone | 1 (flat) + 1 (curved) = 2 surfaces | 1 (curved) | 1 |
| Sphere | 0 (flat) + 1 (curved) = 1 surface | 0 | 0 |
| Square Pyramid | 5 (1 square + 4 triangular) | 8 | 5 |
| Triangular Pyramid (Tetrahedron) | 4 (all triangular) | 6 | 4 |
| Triangular Prism | 5 (2 triangular + 3 rectangular) | 9 | 6 |
Understanding the number of faces, edges, and vertices helps in identifying and describing solid shapes.
Example 1. Identify the shape of a brick and state the number of faces, edges, and vertices it has.
Answer:
A typical brick is in the shape of a Cuboid.
As we listed in the properties of a cuboid:
- Number of Faces: 6
- Number of Edges: 12
- Number of Vertices: 8
So, a brick has 6 flat faces, 12 straight edges, and 8 vertices.
Example 2. What is the shape of the base of a square pyramid? How many edges does this base have?
Answer:
The name "square pyramid" tells us about the shape of its base. The base of a square pyramid is a square.
A square is a 2D shape with 4 sides and 4 vertices. The sides of the base are the edges of the base.
The base of a square pyramid is a square, and a square has 4 sides (edges). Therefore, the base has 4 edges.
Drawing Solids on a Flat Surface
Solid shapes, being three-dimensional (3D), exist in space and have length, breadth, and height. However, we often need to represent or draw these 3D shapes on a two-dimensional (2D) surface, like a sheet of paper or a computer screen, which only have length and breadth. This process is called visualising solid shapes or drawing them in 2D. Since a 2D surface lacks the dimension of depth, we use special techniques to create the illusion of three dimensions and make the drawing look realistic or convey the shape's features accurately.
Representing 3D Shapes in 2D
There are various methods used to draw 3D shapes on a 2D surface. Two common methods that are useful for understanding how to represent solids are:
- Oblique sketches
- Isometric sketches
1. Oblique Sketches:
An oblique sketch is a simple method for drawing solid shapes on a flat surface. In an oblique sketch, some aspects of the solid are drawn in their true shape and size, while others are distorted to show depth.
Key characteristics of an oblique sketch:
- The front face of the solid is typically drawn as its true shape and size. For example, if drawing a cube, the front face will be a perfect square.
- The edges that go back into the distance from the front face (often called receding edges) are drawn at an angle to the horizontal. A common angle used is $45^\circ$, but other angles can also be used.
- To create the illusion of depth, these receding edges are often drawn shorter than their actual length in the real solid. The amount by which they are shortened can vary, making the sketch not perfectly to scale.
- Parallel edges in the solid remain parallel in the oblique sketch.
Oblique sketches are relatively easy to draw because the front face dimensions are kept true. However, because the receding edges are shortened arbitrarily (not to scale), the sketch might not look exactly like the real object and is not suitable for precise measurements.
Steps to draw an Oblique Sketch of a Cuboid (e.g., with dimensions 5 units $\times$ 3 units $\times$ 2 units):
1. Choose one face as the front face. Draw it as a rectangle with its true dimensions, say 5 units wide and 3 units high.
2. From each of the four vertices of the front rectangle, draw a line segment representing the depth (2 units). These lines are the receding edges. Draw them at an angle (e.g., $45^\circ$) to the horizontal edges of the front face. For a sense of depth, you might draw these lines shorter than 2 units, for example, 1 unit or 1.5 units, but label them as representing 2 units. Use dotted lines for edges that would be hidden behind the solid.
3. Join the endpoints of the receding edges to complete the back face of the cuboid. The back face should be parallel to the front face and have the same true dimensions (5 units $\times$ 3 units) in the sketch.
2. Isometric Sketches:
An isometric sketch is another method for drawing 3D shapes on a 2D surface. Unlike oblique sketches, isometric sketches attempt to show all edges in their relative proportions, making them look more realistic and suitable for scale drawings (though scaling is not always required at this level).
Isometric sketches are often drawn on a special type of paper called isometric dot paper or isometric grid paper. This paper has dots arranged in a grid of equilateral triangles.
Key characteristics of an isometric sketch:
- The sketch is based on three axes that meet at a vertex, representing the three dimensions (length, width, height). On isometric paper, two of these axes are typically at $30^\circ$ to the horizontal line of the paper, and the third is vertical ($90^\circ$ to the horizontal). The angle between any two of these axes is $120^\circ$.
- All edges parallel to these axes are drawn to their true length, measured using the distance between the dots on the isometric paper. This is a key difference from oblique sketches where receding edges are often shortened.
- Parallel edges in the solid remain parallel in the isometric sketch.
- Angles in the solid do not necessarily appear as their true measure in the sketch (except for angles between edges parallel to the axes). For example, the right angles of a cube's faces appear as $120^\circ$ or $60^\circ$ in the sketch.
Isometric sketches provide a more realistic representation and are often used in technical drawings.
Steps to draw an Isometric Sketch of a Cube (e.g., with side length 3 units) on Isometric Dot Paper:
1. Identify a starting vertex. Draw a vertical line segment of length 3 units (connecting 3 intervals of dots) representing one vertical edge of the cube.
2. From the bottom endpoint of the vertical edge, draw two line segments, each of length 3 units, along the two slanting lines of the isometric grid. These lines should make angles of $120^\circ$ with the vertical line and represent two edges of the base.
3. From the top endpoint of the vertical edge, draw two line segments, each of length 3 units, parallel to the two base edges drawn in step 2. These represent the top edges meeting at the top vertex.
4. Complete the sketch by drawing the remaining edges. These will connect the endpoints of the lines drawn in step 2 and 3. Ensure these new edges are parallel to the first vertical edge drawn and also have length 3 units. Use dotted lines for edges that would be hidden.
Comparing Oblique and Isometric Sketches
Let's summarise the main differences:
| Feature | Oblique Sketch | Isometric Sketch |
|---|---|---|
| Front Face | True shape and size | Shape and size may be distorted (not true), but edges parallel to axes are in proportion |
| Receding Edges | Drawn shorter than true length (arbitrary shortening) | Drawn to true length (proportional) |
| Angle of Receding Edges | Usually $45^\circ$ to horizontal (can be other angles) | Two axes at $30^\circ$ to horizontal, one vertical ($90^\circ$); angles between axes are $120^\circ$ |
| Parallelism | Parallel edges remain parallel | Parallel edges remain parallel |
| Realism | Less realistic | More realistic |
| Paper Used | Regular plain or grid paper | Often uses isometric dot/grid paper |
Example 1. Draw an oblique sketch of a cuboid with dimensions 4 cm (length), 3 cm (breadth), and 2 cm (height).
Answer:
We will draw the front face as a rectangle measuring 4 cm wide and 3 cm high. Then, from each vertex, we will draw receding edges at an angle (say $45^\circ$) representing the 2 cm depth. For a good visual effect, we can draw the receding edges shorter, perhaps about half the depth, i.e., 1 cm long, but understand they represent a depth of 2 cm.
In the sketch, the front rectangle shows the true 4 cm $\times$ 3 cm face. The lines going back represent the 2 cm depth, drawn at an angle (around $45^\circ$) and typically shortened to improve the visual appearance on a flat surface.
Example 2. Draw an isometric sketch of a cuboid of dimensions 3 units $\times$ 2 units $\times$ 1 unit on isometric dot paper.
Answer:
We need to draw a cuboid using the isometric grid, ensuring that edges parallel to the three isometric axes are drawn to their true lengths of 3, 2, and 1 unit.
1. Start by choosing a vertex. Let's draw the vertical edge first, of length 1 unit (connecting one interval of dots vertically).
2. From the bottom of this vertical edge, draw an edge of length 3 units along one slanting direction and an edge of length 2 units along the other slanting direction.
3. From the top of the vertical edge, draw lines parallel to the 3-unit and 2-unit edges drawn in step 2, each of the same respective lengths.
4. Connect the endpoints to complete the cuboid. All edges should follow the isometric grid lines and be parallel to their corresponding edges.
In this isometric sketch, notice that the lengths of 3, 2, and 1 units are represented accurately along the directions of the isometric axes using the dot grid.
Visualising Different Sections of a Solid
In the previous sections, we learned about different solid shapes and how to draw them on a flat surface. Now, we will explore what happens when we cut through a solid shape. This helps us understand the internal structure or the shape of the boundary created by the cut. The 2D shape obtained by cutting through a solid with a plane is called a cross-section.
Understanding Cross-Sections
Imagine you have a solid object, like a loaf of bread, and you slice it with a knife. The flat surface that is revealed on the bread where you made the cut is a cross-section. A cross-section is essentially the intersection of a 3D solid shape with a 2D plane (like the plane of the knife blade).
The shape of the cross-section depends on two main things:
- The type of solid shape being cut.
- The angle and position of the cutting plane.
Methods of Cutting:
We can slice a solid shape in different ways. Two common ways to understand cross-sections at a basic level are:
- Vertical Cut: This is a cut made perpendicular to the base of the solid. Imagine slicing a standing object straight down from top to bottom.
- Horizontal Cut: This is a cut made parallel to the base of the solid. Imagine slicing across the object, parallel to the surface it is resting on.
Cuts can also be made at other angles, leading to a wider variety of cross-sections.
Examples of Cross-Sections of Common Solids
Let's look at the cross-sections obtained by making simple horizontal and vertical cuts through some familiar solid shapes.
Cube:
Vertical or Horizontal Cut parallel to a face:
When a cube is cut vertically or horizontally with a plane parallel to one of its faces, the cross-section obtained is a square. This is because all faces of a cube are squares, and a parallel cut reveals a congruent square face.Diagonal Cut:
If you cut a cube diagonally (not parallel to a face), you can get other shapes. For example, a cut through diagonally opposite vertices might give a rectangle. A cut passing through the midpoints of several edges can even result in a hexagon.
Cylinder:
Vertical Cut (through the center):
When a cylinder standing on its circular base is cut vertically by a plane passing through the center of the bases, the cross-section obtained is a rectangle. The height of the rectangle is the height of the cylinder, and the width is the diameter of the base.Horizontal Cut:
When a cylinder is cut horizontally by a plane parallel to its circular bases, the cross-section obtained is a circle. All horizontal cross-sections of a cylinder (except possibly at the very top or bottom edge) will be circles identical to the base.
Cone:
Vertical Cut (through the apex):
When a cone standing on its circular base is cut vertically by a plane passing through its apex and the center of its base, the cross-section obtained is a triangle. Specifically, it's an isosceles triangle if the cone is a right cone.Horizontal Cut:
When a cone is cut horizontally by a plane parallel to its circular base, the cross-section obtained is a circle. As the horizontal cut is made closer to the apex, the circle gets smaller.
Sphere:
Any Cut:
When a sphere is cut by any plane (vertical, horizontal, or at any angle), the cross-section obtained is always a circle. If the cutting plane passes through the center of the sphere, the resulting circle is the largest possible cross-section, called a great circle. If the cut is not through the center, a smaller circle is obtained.
Example 1. What cross-section do you get when you give a vertical cut to a die (which is a cube)?
Answer:
A die is a small cube used in games. When you make a vertical cut through a cube, parallel to one of its faces, the cutting plane intersects the solid to form a 2D shape. Since the cube has square faces and the cut is parallel to a face, the resulting shape will be a square that is congruent to the faces of the cube.
Thus, a vertical cut parallel to a face of a die (cube) results in a square cross-section.
Example 2. Describe the cross-sections obtained by a horizontal cut and a vertical cut (passing through the center) of a cylinder.
Answer:
Let's consider a cylinder standing upright on its base.
Horizontal Cut: A horizontal cut is made by a plane parallel to the base of the cylinder. Since the base is a circle and the cylinder's sides are straight up, slicing parallel to the base at any height will reveal a circular surface.
So, a horizontal cut of a cylinder results in a circle.
Vertical Cut (through the center): A vertical cut passing through the center means the cutting plane contains the axis of the cylinder (the line joining the centers of the two circular bases). This plane slices through the circular bases along their diameters and cuts the curved surface along two parallel lines (the generators of the cylinder). The resulting shape has two vertical sides (from the curved surface) and two horizontal sides (from the diameters of the bases).
So, a vertical cut through the center of a cylinder results in a rectangle.