Visualization of Solid Figures — Questions and Answers
Welcome to HSLC Guru. This page gives complete, step-by-step solutions for ASSEB Class 8 General Mathematics Chapter 10, Visualization of Solid Figures — every Exercise, in-text Activity and extra question. It covers front, side and top views of 3D objects, faces, edges and vertices of polyhedrons, Euler’s formula and drawing maps, all explained with clear labelled figures.
Summary
Objects around us are of two kinds — plane (two-dimensional, 2D) figures such as triangle, circle, square, rectangle and quadrilateral, and solid (three-dimensional, 3D) figures such as cube, rectangular parallelopiped, cylinder, cone, sphere, pyramid and prism. A 3D object looks different when viewed from different positions, giving a front view, a side view and a top view.
Solid objects made only of flat polygonal faces are called polyhedrons. For every polyhedron the number of faces (F), vertices (V) and edges (E) are related by Euler’s formula $F + V – E = 2$. Sphere, cylinder and cone are not polyhedrons because they have curved surfaces.
A map shows the position of one place relative to another. A map must be drawn with a proper scale so that the distances shown are proportional to the actual distances. Unlike a picture, a map looks the same from whatever position the observer views it.
Summary: This ASSEB Class 8 General Mathematics Chapter 10 (Visualization of Solid Figures) guide gives fully worked answers to Exercise 10.1 and Exercise 10.2 and every in-text activity. It covers front, side and top views of 3D objects, faces, edges and vertices of polyhedrons, Euler’s formula F + V − E = 2, and drawing maps with a proper scale, all with clear labelled figures.
Textbook Questions and Answers
Activity: Match the figure, name and dimension
Question: Match the dimension and name of the shape with the figures of the given shapes.
Answer: The correct matches are given below. Square and Quadrilateral are plane (2D); all the others are solid (3D).
| Name of the shape | Dimension |
|---|---|
| Square | Two dimensional (2D) |
| Quadrilateral | Two dimensional (2D) |
| Cone | Three dimensional (3D) |
| Cylinder | Three dimensional (3D) |
| Rectangular parallelopiped | Three dimensional (3D) |
| Sphere | Three dimensional (3D) |
10.1 View of three-dimensional figures from different positions
When a house is seen from the front we get the front view, from the side the side view, and from above the top view. The same object looks different from different positions.
Exercise 10.1
1. Some three-dimensional objects are given below. The front, top and side views of the objects are also given. For each object, identify which drawing is the front, top and side view.
Answer: The view in which most detail (lock, handle, screen, etc.) is seen is the front view; the flat drawing obtained by looking from above is the top view; the drawing seen from the side is the side view. The identification for each object is given below.
(a) Box (trunk):
Answer: (i) the drawing with the lock in the middle is the front view; (ii) the drawing with the side handle and top hinges is the side view; (iii) the plain rectangle with no detail is the top view (the lid seen from above).
(b) Almirah:
Answer: (i) the wide drawing with doors, mirror and handle is the front view; (ii) the tall, narrow rectangle is the side view (the almirah is shallow, so its side view is narrow); (iii) the long, flat rectangle is the top view.
(c) Book:
Answer: (i) the cover marked ‘BOOK’ is the front view; (ii) and (iii) are the two thin rectangles — because a book is thin, both its top view and side view appear as thin rectangles.
(d) Brick:
Answer: A brick is a rectangular parallelopiped, so all three views are rectangles. (i) the length × height rectangle is the front view; (ii) the largest length × breadth rectangle is the top view; (iii) the small, narrow rectangle (breadth × height) is the side view.
(e) TV:
Answer: (i) the drawing showing the screen is the front view; (ii) the trapezium — wide at the front and narrowing towards the back — is the top view; (iii) the drawing showing the depth and the speaker holes is the side view.
(f) Weight:
Answer: (i) is the pictorial (3-D) drawing of the weight; (ii) the hexagon marked ‘1kg’ is the top view (from above it looks like a hexagon); (iii) the trapezium is the front view. As the weight is hexagonal, its side view is also the same trapezium.
(g) House:
Answer: (i) the slanted drawing is the pictorial (3-D) figure of the house; (ii) the narrow gable end with the triangular roof is the side view; (iii) the wide face with the door and windows is the front view; (iv) the roof seen from above is the top view.
(h) Dustbin:
Answer: (i) the bucket shape (wide at the top, narrow at the bottom) is the front view; (ii) the pair of concentric circles is the top view. As the dustbin is cylindrical, its front and side views are the same, so only two views are drawn.
Activity: Draw the views
Question: By discussing in a group, draw the front, side and top views of the following solid objects — (i) a chalk pencil box (ii) the almirah in your school (iii) the school building (iv) a table and desk-bench in your classroom.
Answer: This is a hands-on activity. The views come out as follows — Chalk box: all three views are rectangles (it is a rectangular parallelopiped). Almirah: front view a wide rectangle (with doors), side view a tall narrow rectangle, top view a flat rectangle. School building: like the house above — door and windows in the front, a triangular gable on the side, the roof rectangle on top. Table / desk-bench: the top view is a rectangle, while the legs appear in the front and side views.
10.2 Faces, Edges and Vertices
Every solid object occupies space. Some solids are made of flat polygonal faces — such solids are called polyhedrons (‘Poly’ means many, ‘Hedron’ means base or face). A flat face of a polyhedron is a face (F), the straight line where two faces meet is an edge (E), and the point where edges meet is a vertex (V).
The table below gives the number of vertices (V), faces (F) and edges (E) of a cube, rectangular parallelopiped, cylinder, sphere and cone.
| Solid object | Vertices (V) | Faces (F) | Edges (E) |
|---|---|---|---|
| Cube | 8 | 6 | 12 |
| Rectangular parallelopiped | 8 | 6 | 12 |
| Cylinder | 0 | 2 flat + 1 curved | 2 |
| Sphere | 0 | 1 curved | 0 |
| Cone | 1 | 1 flat + 1 curved | 1 |
Sphere, cylinder and cone are not polyhedrons because they have curved surfaces and their base is not a polygon. A polyhedron may be regular or irregular. In a regular polyhedron every face is a congruent regular polygon and every vertex is formed by the same number of faces — for example the cube, the regular tetrahedron and the regular octahedron. A rectangular parallelopiped or a square-based pyramid is an irregular polyhedron because its faces are not congruent.
Euler’s formula (completing the table)
On computing F, V and E for the polyhedrons below, the values of F+V and E+2 always come out equal. The completed table is shown here.
| Polyhedron | F | V | E | F+V | E+2 |
|---|---|---|---|---|---|
| Cube | 6 | 8 | 12 | 14 | 14 |
| Rectangular parallelopiped | 6 | 8 | 12 | 14 | 14 |
| Pyramid with square base | 5 | 5 | 8 | 10 | 10 |
| Prism with square base | 6 | 8 | 12 | 14 | 14 |
| Prism with triangular base | 5 | 6 | 9 | 11 | 11 |
| Pyramid with triangular base | 4 | 4 | 6 | 8 | 8 |
In every row F+V equals E+2. Hence, for any polyhedron Euler’s formula is—
$$F + V – E = 2$$
This relation was discovered by the great mathematician Leonhard Euler (1707–1783) and is known as Euler’s formula for polyhedra.
Try yourself
Question: Take a cube. On cutting one corner of it you get a new shape. Find F, V and E of this shape and verify Euler’s formula. Also, is a prism with a square base similar to a cube in any way?
Answer: When one corner (one vertex) of a cube is cut, a new triangular face appears there. So one face is added ($F: 6 \to 7$), the one vertex is replaced by three new vertices ($V: 8 \to 10$) and three new edges are added ($E: 12 \to 15$). Check: $F + V – E = 7 + 10 – 15 = 2$ — Euler’s formula holds. Again, a prism with a square base has F=6, V=8, E=12 — exactly the same as a cube; so a cube is in fact a special square prism.
Exercise 10.2
1. For a polyhedron, the number of vertices and faces are 12 and 20 respectively. Find the number of edges.
Answer: Here $V = 12$ and $F = 20$. Using Euler’s formula $F + V – E = 2$—
$$E = F + V – 2 = 20 + 12 – 2 = 30$$
So the number of edges is 30.
2. For a polyhedron, the number of edges and vertices are 12 and 6 respectively. Find the number of faces.
Answer: Here $E = 12$ and $V = 6$. From Euler’s formula—
$$F = E – V + 2 = 12 – 6 + 2 = 8$$
So the number of faces is 8.
3. Verify Euler’s formula for the two polyhedrons given below (a pyramid with a triangular base and a prism with a square base).
Answer: For the pyramid with a triangular base (a tetrahedron), $F = 4$, $V = 4$, $E = 6$—
$$F + V – E = 4 + 4 – 6 = 2$$
For the prism with a square base, $F = 6$, $V = 8$, $E = 12$—
$$F + V – E = 6 + 8 – 12 = 2$$
In both cases $F + V – E = 2$. Hence Euler’s formula is verified for both polyhedrons.
10.3 Drawing a map of our surrounding places
A map tells us how to travel from one place to another and how far it is. Different symbols and colours are used to show different places, rivers and hills — green for forests, blue for rivers, and so on. A map must be drawn with a proper scale so that the distances shown are proportional to the actual distances.
In the map above, Kayum has drawn the way from her house to her friend Sristi’s house, using different symbols for the library, school, market, hospital, community hall, bihutali and other places on the way. She used the scale 1 cm = 100 m of actual distance. Two maps may be the same size but have different scales; for example, the maps of Assam and India may be the same size, but the actual distance shown by 1 cm on the India map is greater than that on the Assam map.
Activity: Like Kayum, draw a map of the road from your house to your school, using symbols for the places on the road and a suitable scale. (This is a hands-on activity.)
Additional Questions and Answers
Multiple Choice Questions (MCQ)
1. How many edges does a cube have? (a) 6 (b) 8 (c) 12 (d) 4
Answer: (c) 12.
2. Euler’s formula is— (a) $F + V + E = 2$ (b) $F + V – E = 2$ (c) $F – V + E = 2$ (d) $F + V – E = 0$
Answer: (b) $F + V – E = 2$.
3. The number of vertices of a sphere is— (a) 0 (b) 1 (c) 2 (d) 4
Answer: (a) 0.
4. Which of the following is not a polyhedron? (a) Cube (b) Pyramid (c) Prism (d) Sphere
Answer: (d) Sphere (because it has a curved surface).
5. The number of vertices of a cone is— (a) 0 (b) 1 (c) 2 (d) 3
Answer: (b) 1.
6. The number of faces of a pyramid with a triangular base is— (a) 3 (b) 4 (c) 5 (d) 6
Answer: (b) 4.
7. The number of vertices of a pyramid with a square base is— (a) 4 (b) 5 (c) 6 (d) 8
Answer: (b) 5.
8. Which colour is usually used to show a forest on a map? (a) Blue (b) Green (c) Red (d) Yellow
Answer: (b) Green.
9. Each vertex of a cube is formed by how many faces? (a) 2 (b) 3 (c) 4 (d) 6
Answer: (b) 3.
10. For a polyhedron $V = 6$ and $F = 8$; the number of edges is— (a) 10 (b) 12 (c) 14 (d) 16
Answer: (b) 12, because $E = F + V – 2 = 8 + 6 – 2 = 12$.
Fill in the blanks
1. A three-dimensional object ______ space.
Answer: occupies.
2. Euler’s formula: $F + V – E =$ ______.
Answer: $2$.
3. A solid made of flat polygonal faces is called a ______.
Answer: polyhedron.
4. The number of vertices of a cube is ______.
Answer: 8.
5. On a map, ______ colour is used to show a river.
Answer: blue.
True or False
1. A sphere is a polyhedron.
Answer: False (a sphere has a curved surface).
2. A cone has one vertex.
Answer: True.
3. A rectangular parallelopiped is a regular polyhedron.
Answer: False (its faces are not congruent, so it is an irregular polyhedron).
4. A scale is needed to draw a map.
Answer: True.
5. A three-dimensional object looks the same from every position.
Answer: False (it looks different from different positions).
Short Answer Questions
1. What is a polyhedron? Write its definition.
Answer: A solid object made only of flat polygonal faces, whose edges are straight lines, is called a polyhedron. Examples — cube, rectangular parallelopiped, prism, pyramid.
2. What is the difference between a regular and an irregular polyhedron?
Answer: In a regular polyhedron every face is a congruent regular polygon and every vertex is formed by the same number of faces (for example, a cube or a regular tetrahedron). In an irregular polyhedron the faces are not congruent (for example, a rectangular parallelopiped or a square-based pyramid).
3. What is the difference between drawing a map and drawing a picture?
Answer: A picture is drawn as we see it, so it looks different when the observer’s position changes. A map, however, stays the same from whatever position it is viewed; it is always a top view drawn to a fixed scale.
4. A polyhedron has $F = 8$ and $E = 12$. Find $V$ using Euler’s formula.
Answer: $V = E – F + 2 = 12 – 8 + 2 = 6$. So the number of vertices is 6.
Key Terms
| Term | Meaning |
|---|---|
| Solid figure | A three-dimensional object having length, breadth and height |
| Front view | The shape seen from the front |
| Side view | The shape seen from the side |
| Top view | The shape seen from above |
| Polyhedron | A solid made of flat polygonal faces |
| Face | A flat surface of a polyhedron |
| Edge | The straight line where two faces meet |
| Vertex | The point where edges meet |
| Euler’s formula | $F + V – E = 2$ for any polyhedron |
| Map | A scaled top-view drawing of a place |
| Scale | The ratio of map distance to actual distance |