1
Below are given some 3D-solids and their three different views. Label the views shown as front view, side view and top view and tick the one which is the best 2D-representation of the object shown:
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Three-Dimensional Solids
ICSE Class 8 Mathematics · Solutions for Class 8 RS Aggarwal ICSE Foundation Mathematics · 37 solved questions
2
Complete the following table by filling the number of faces, the number of edges and the number of vertices for the given solid:
| Solid | Number of Faces | Number of Edges | Number of Vertices |
|---|---|---|---|
| (i) Cuboid | ...... | ...... | ...... |
| (ii) Tetrahedron | ...... | ...... | ...... |
| (iii) Rectangular Pyramid | ...... | ...... | ...... |
| (iv) Pentagonal Pyramid | ...... | ...... | ...... |
| (v) Hexagonal Pyramid | ...... | ...... | ...... |
| (vi) Triangular Prism | ...... | ...... | ...... |
| (vii) Hexagonal Prism | ...... | ...... | ...... |
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(i) A cuboid is bounded by 6 rectangular faces. So, it has 6 faces, 12 edges and 8 vertices.
(ii) A tetrahedron is a triangular pyramid. It has 1 triangular base and 3 triangular lateral faces. So, it has 4 faces, 6 edges and 4 vertices.
(iii) A rectangular pyramid has 1 rectangular base and 4 triangular lateral faces. So, it has 5 faces, 8 edges and 5 vertices.
(iv) A pentagonal pyramid has 1 pentagonal base and 5 triangular lateral faces. So, it has 6 faces, 10 edges and 6 vertices.
(v) A hexagonal pyramid has 1 hexagonal base and 6 triangular lateral faces. So, it has 7 faces, 12 edges and 7 vertices.
(vi) A triangular prism has 2 triangular faces and 3 rectangular lateral faces. So, it has 5 faces, 9 edges and 6 vertices.
(vii) A hexagonal prism has 2 hexagonal faces and 6 rectangular lateral faces. So, it has 8 faces, 18 edges and 12 vertices.
The completed table is shown below:
| Solid | Number of Faces | Number of Edges | Number of Vertices |
|---|---|---|---|
| (i) Cuboid | 6 | 12 | 8 |
| (ii) Tetrahedron | 4 | 6 | 4 |
| (iii) Rectangular Pyramid | 5 | 8 | 5 |
| (iv) Pentagonal Pyramid | 6 | 10 | 6 |
| (v) Hexagonal Pyramid | 7 | 12 | 7 |
| (vi) Triangular Prism | 5 | 9 | 6 |
| (vii) Hexagonal Prism | 8 | 18 | 12 |
3
Fill in the blanks:
(i) A square prism may also be called a ......................
(ii) A tetrahedron has ...................... triangular faces.
(iii) A pentagonal pyramid has ...................... triangular faces.
(iv) The five lateral faces of a pentagonal prism are ...................... .
(v) Two adjacent faces of a solid meet in a/an ......................
(vi) Two adjacent edges of a solid meet in a/an ......................
(vii) The side view of any pyramid depicts a ......................
(viii) If a prism has a $n$-sided polygon as its base, then it has ............ faces ............ edges and ............ vertices.
(ix) If a pyramid has a $n$-sided polygon as its base, then it has ............ faces, ............ edges and ............ vertices.
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(i) A square prism may also be called a cuboid.
(ii) A tetrahedron has 4 triangular faces.
(iii) A pentagonal pyramid has 5 triangular faces.
(iv) The five lateral faces of a pentagonal prism are rectangles.
(v) Two adjacent faces of a solid meet in a/an edge.
(vi) Two adjacent edges of a solid meet in a/an vertex.
(vii) The side view of any pyramid depicts a triangle.
(viii) If a prism has a $n$-sided polygon as its base, then it has $\bm{n + 2}$ faces $\bm{3n}$ edges and $\bm{2n}$ vertices.
(ix) If a pyramid has a $n$-sided polygon as its base, then it has $\bm{n + 1}$ faces, $\bm{2n}$ edges and $\bm{n + 1}$ vertices.
4
Write the number of faces, edges and vertices in:
(i) an octagonal pyramid
(ii) a decagonal prism
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(i) In an octagonal pyramid, the number of sides of the polygonal base is 8.
Number of faces in a pyramid is 1 more than the number of sides of the polygonal base.
Number of faces = 8 + 1 = 9
Number of edges in a pyramid is twice the number of sides of the polygonal base.
Number of edges = 8 × 2 = 16
Number of vertices in a pyramid is 1 more than the number of sides of the polygonal base.
Number of vertices = 8 + 1 = 9
Hence, in an octagonal pyramid, F = 9, E = 16 and V = 9.
(ii) In a decagonal prism, the number of sides of the polygonal base is 10.
Number of faces in a prism is 2 more than the number of sides of the polygonal base.
Number of faces = 10 + 2 = 12
Number of edges in a prism is thrice the number of sides of the polygonal base.
Number of edges = 10 × 3 = 30
Number of vertices in a prism is twice the number of sides of the polygonal base.
Number of vertices = 10 × 2 = 20
Hence, in a decagonal prism, F = 12, E = 30 and V = 20.
5
Which of the following solids has the highest number of vertices?
(i) Cuboid
(ii) Triangular Prism
(iii) Triangular Pyramid
(iv) Square Pyramid
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Number of vertices in a cuboid = 8
Number of vertices in a triangular prism = 6
Number of vertices in a triangular pyramid = 4
Number of vertices in a square pyramid = 5
Of these, 8 is the greatest.
Hence, cuboid has the highest number of vertices.
6
In which of the following solids, all the lateral faces are not triangular?
(i) Triangular Prism
(ii) Triangular Pyramid
(iii) Square Pyramid
(iv) Pentagonal Pyramid
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A triangular pyramid, a square pyramid and a pentagonal pyramid are all pyramids, and all the lateral faces of a pyramid are triangles.
A triangular prism is a prism, and its 3 lateral faces are rectangles.
Hence, in a triangular prism all the lateral faces are not triangular.
7
In which of the following solids, none of the faces is triangular?
(i) Triangular Prism
(ii) Square Pyramid
(iii) Pentagonal Pyramid
(iv) Hexagonal Prism
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A triangular prism has 2 triangular faces and 3 rectangular faces.
A square pyramid has 1 square face and 4 triangular faces.
A pentagonal pyramid has 1 pentagonal face and 5 triangular faces.
A hexagonal prism has 2 hexagonal faces and 6 rectangular faces.
Hence, in a hexagonal prism, none of the faces is triangular.
8
Look at the following picture:
Which of the options given below correctly represent the view of the above object as seen from the top?
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The given object is a container in the shape of a cuboid, having an oval cap fitted on its upper face.
Looking at the object from the top, the rectangular upper face of the container is seen along with the oval cap lying on it.
So, the top view of the given object is a rectangle having an oval inside it, which is the figure given in option (ii).
Hence, option (ii) is the correct option.
9
Choose the correct top view of the following jointed figure from the options given below it:
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The given jointed figure is made of 6 identical cubes. One cube is at the centre, four cubes are joined to the four side faces of this central cube and one cube is placed on the top of the central cube.
Looking at the figure from the top, the central cube and the four cubes joined to its side faces are seen as 5 squares forming the shape of a plus sign. The cube placed on the top of the central cube exactly covers the central square, so it does not change the shape seen.
So, the top view of the given jointed figure is the figure given in option (iii).
Hence, option (iii) is the correct option.
10
Below are given three views of a 3D-jointed figure:
Choose the correct corresponding figure:
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In the front view, there are 3 squares in a row with 2 more squares placed one above the other on the middle square. So, the required figure is 3 cubes long and 3 cubes high, and the column of cubes stands on the middle cube.
In the top view, there are 3 squares in a row. So, the required figure is 3 cubes long and 1 cube wide.
In the side view, there are 3 squares placed one above the other. So, the required figure is 1 cube wide and 3 cubes high.
The figure given in option (iii) is made of 3 cubes placed in a row with 2 cubes placed one above the other on the middle cube, so it has all the three given views.
Hence, option (iii) is the correct option.
11
Draw the correct 2D-front view of the jointed figure shown below:
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The given jointed figure is made of 4 identical cubes. Three cubes are placed in a row at the back and one cube is joined in front of the middle cube of this row.
Looking at the figure from the front, the cube lying in front exactly covers the middle cube of the row behind it.
So, only 3 squares placed in a row are seen.
The required 2D-front view of the given jointed figure is shown below:
1(i)
Verify Euler's relation for a cuboid.
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In a cuboid, we have
Number of faces = 6
Number of edges = 12
Number of vertices = 8
∴ F = 6, E = 12 and V = 8
∴ (F - E + V) = (6 - 12 + 8) = 2
Hence, Euler's relation is verified.
1(ii)
Verify Euler's relation for a triangular prism.
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In a triangular prism, we have
Number of faces = 2 triangular + 3 rectangular = 5
Number of edges = 9
Number of vertices = 6
∴ F = 5, E = 9 and V = 6
∴ (F - E + V) = (5 - 9 + 6) = 2
Hence, Euler's relation is verified.
1(iii)
Verify Euler's relation for a square pyramid.
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In a square pyramid, we have
Number of faces = 1 square + 4 triangular = 5
Number of edges = 8
Number of vertices = 5
∴ F = 5, E = 8 and V = 5
∴ (F - E + V) = (5 - 8 + 5) = 2
Hence, Euler's relation is verified.
1(iv)
Verify Euler's relation for a tetrahedron.
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In a tetrahedron, we have
Number of faces = 4
Number of edges = 6
Number of vertices = 4
∴ F = 4, E = 6 and V = 4
∴ (F - E + V) = (4 - 6 + 4) = 2
Hence, Euler's relation is verified.
2(i)
Verify Euler's relation for the following solid:
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The given solid is an octahedron.
From the figure,
Number of faces = 8
Number of edges = 12
Number of vertices = 6
∴ F = 8, E = 12 and V = 6
∴ (F - E + V) = (8 - 12 + 6) = 2
Hence, Euler's relation is verified.
2(ii)
Verify Euler's relation for the following solid:
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The given solid is a hexagonal prism.
From the figure,
Number of faces = 2 hexagonal + 6 rectangular = 8
Number of edges = 18
Number of vertices = 12
∴ F = 8, E = 18 and V = 12
∴ (F - E + V) = (8 - 18 + 12) = 2
Hence, Euler's relation is verified.
3
If a polyhedron has 8 faces and 10 vertices, find the number of edges.
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Given:
F = 8, V = 10
According to Euler's relation,
F - E + V = 2
⇒ 8 - E + 10 = 2
⇒ 18 - E = 2
⇒ E = 18 - 2
⇒ E = 16
Hence, the number of edges = 16.
4
If a polyhedron has 8 faces and 16 edges, find the number of vertices.
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Given:
F = 8, E = 16
According to Euler's relation,
F - E + V = 2
⇒ 8 - 16 + V = 2
⇒ V - 8 = 2
⇒ V = 2 + 8
⇒ V = 10
Hence, the number of vertices = 10.
5(i)
Write the number of faces, vertices and edges in an octagonal pyramid and verify Euler's relation for the same.
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In an octagonal pyramid, the number of sides of the polygonal base is 8.
Number of faces in a pyramid is 1 more than the number of sides of the polygonal base.
Number of faces = 8 + 1 = 9
Number of vertices in a pyramid is 1 more than the number of sides of the polygonal base.
Number of vertices = 8 + 1 = 9
Number of edges in a pyramid is twice the number of sides of the polygonal base.
Number of edges = 8 × 2 = 16
∴ F = 9, E = 16 and V = 9
∴ (F - E + V) = (9 - 16 + 9) = 2
Hence, in an octagonal pyramid, F = 9, V = 9, E = 16 and Euler's relation is verified.
5(ii)
Write the number of faces, vertices and edges in a decagonal prism and verify Euler's relation for the same.
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In a decagonal prism, the number of sides of the polygonal base is 10.
Number of faces in a prism is 2 more than the number of sides of the polygonal base.
Number of faces = 10 + 2 = 12
Number of vertices in a prism is twice the number of sides of the polygonal base.
Number of vertices = 10 × 2 = 20
Number of edges in a prism is thrice the number of sides of the polygonal base.
Number of edges = 10 × 3 = 30
∴ F = 12, E = 30 and V = 20
∴ (F - E + V) = (12 - 30 + 20) = 2
Hence, in a decagonal prism, F = 12, V = 20, E = 30 and Euler's relation is verified.
1(i)
Which solid figure would the following net make when folded along dotted lines?
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The given net consists of 6 rectangles. Four of these rectangles are placed one below the other forming a vertical strip, and the remaining two rectangles are attached to the two sides of the second rectangle of this strip.
On folding the net along the dotted lines, the four rectangles of the vertical strip form the four side faces of the solid, and the two rectangles attached to the sides form its remaining two opposite faces.
So, the solid formed is bounded by 6 rectangular faces.
Hence, the given net makes a cuboid.
1(ii)
Which solid figure would the following net make when folded along dotted lines?
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The given net consists of 3 rectangles placed one below the other forming a vertical strip, and 2 triangles attached to the two sides of the middle rectangle of this strip.
On folding the net along the dotted lines, the 3 rectangles form the 3 lateral faces of the solid, and the 2 triangles form its 2 congruent triangular faces at the two ends.
So, the solid formed has 2 congruent triangular faces and 3 rectangular lateral faces.
Hence, the given net makes a triangular prism.
2
Which of these nets cannot be folded to form a cube?
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A net can be folded to form a cube only if its 6 squares form the 6 faces of the cube, no two of them falling one upon the other.
In net (i), the 4 squares of the upper row form the 4 side faces of the cube. Of these, the second square and the fourth square become opposite faces. So, the 2 squares hanging below them fall one upon the other on the bottom face, leaving one face of the cube open. So, net (i) cannot be folded to form a cube.
In net (iv), the 4 squares of the lower row form the 4 side faces of the cube and the square just above the last of them forms the top face. The topmost square then folds over the top face and falls upon one of the side faces already formed, leaving one face of the cube open. So, net (iv) cannot be folded to form a cube.
In net (vi), the 6 squares form a rectangle of 2 rows and 3 columns. On folding, the squares fall one upon the other and only 4 faces of the cube are formed, leaving 2 faces of the cube open. So, net (vi) cannot be folded to form a cube.
In net (x), the 3 squares of the lower row form 3 faces of the cube, of which the first square and the third square become opposite faces. So, the 2 squares standing above them fall one upon the other, leaving one face of the cube open. So, net (x) cannot be folded to form a cube.
Each of the remaining nets, that is (ii), (iii), (v), (vii), (viii), (ix) and (xi), folds to form a cube.
Hence, the nets given in (i), (iv), (vi) and (x) cannot be folded to form a cube.
3
Which of the following four nets cannot be folded to form a square pyramid?
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A net can be folded to form a square pyramid only if, on folding, a triangle rises from each of the four sides of the square, so that the four triangles meet at a common vertex.
In net (i), one triangle is attached to the upper side of the square and another to its lower side. Each of the remaining two triangles is attached to a slant side of these triangles, so on folding they come to rest on the left side and the right side of the square. So, net (i) can be folded to form a square pyramid.
In net (ii), one triangle is attached to the upper side of the square and another to its lower side. Of the remaining two triangles, one comes to rest on the right side of the square and the other falls upon the triangle standing on the lower side of the square. So, two triangles fall one upon the other and no triangle rises from the left side of the square. So, net (ii) cannot be folded to form a square pyramid.
In net (iii), one triangle is attached to each of the four sides of the square. So, net (iii) can be folded to form a square pyramid.
In net (iv), one triangle is attached to the lower side of the square and another to its right side. Each of the remaining two triangles is attached to a slant side of the triangle on the right, so on folding they come to rest on the upper side and the left side of the square. So, net (iv) can be folded to form a square pyramid.
Hence, the net given in (ii) cannot be folded to form a square pyramid.
4
Shown below is a cuboid.
Draw a net for the cuboid on the square grid given below. Each square is 1 cm × 1 cm.
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A cuboid is bounded by 6 rectangular faces, which form 3 pairs of equal and opposite faces.
For a cuboid of length 4 cm, breadth 3 cm and height 2 cm, we have:
The top face and the bottom face, each of dimensions 4 cm × 3 cm
The front face and the back face, each of dimensions 4 cm × 2 cm
The left face and the right face, each of dimensions 3 cm × 2 cm
On the given square grid, draw the top face, the front face, the bottom face and the back face one below the other forming a vertical strip, each of these rectangles being 4 cm wide. Then draw the left face and the right face on the two sides of the front face.
The required net of the cuboid is shown below:
1
Assertion (A): A polyhedron can have 10 faces, 20 edges and 15 vertices.
Reason (R): For a polyhedron, we have F + V - E = 2.
- (a)Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
- (b)Both Assertion (A) and Reason (R) are true and Reason (R) is not the correct explanation of Assertion (A).
- (c)Assertion (A) is true but Reason (R) is false.
- (d)Assertion (A) is false but Reason (R) is true.
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Given:
F = 10, E = 20, V = 15
For a polyhedron,
F + V - E = 2
L.H.S. = F + V - E = 10 + 15 - 20 = 5
R.H.S. = 2
Since, L.H.S. ≠ R.H.S.
So, a polyhedron cannot have 10 faces, 20 edges and 15 vertices.
So, Assertion (A) is false.
Euler's relation states that in a polyhedron F - E + V = 2, which can also be written as F + V - E = 2.
So, Reason (R) is true.
Hence, option 4 is the correct option.
2
Assertion (A): A regular octahedron has 8 congruent faces which are equilateral triangles.
Reason (R): The net of a 3D shape is a figure in a plane which can be folded to form that 3D shape.
- (a)Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
- (b)Both Assertion (A) and Reason (R) are true and Reason (R) is not the correct explanation of Assertion (A).
- (c)Assertion (A) is true but Reason (R) is false.
- (d)Assertion (A) is false but Reason (R) is true.
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A regular octahedron is a polyhedron bounded by 8 congruent faces, each of which is an equilateral triangle.
So, Assertion (A) is true.
The net of a 3D shape is the figure in a plane which can be folded to form that 3D shape.
So, Reason (R) is true.
Reason (R) tells us what the net of a 3D shape is, which does not explain the number of faces of a regular octahedron or their shape.
So, Reason (R) is not the correct explanation of Assertion (A).
Hence, option 2 is the correct option.
1
The minimum number of faces needed to form a polyhedron is:
- (a)1
- (b)2
- (c)3
- (d)4
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A polyhedron is a solid made up of polygonal regions. The polygon having the least number of sides is a triangle.
Three triangles having a common vertex do not enclose any space, as the solid so formed remains open at its base. So, one more triangular face is needed to close it, and the solid so formed is a tetrahedron.
So, the minimum number of faces needed to form a polyhedron = 4
Hence, option 4 is the correct option.
2
How many minimum faces do you need to form a prism?
- (a)5
- (b)4
- (c)3
- (d)2
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A prism has 2 congruent polygons as its base and its top, and 1 rectangular lateral face along each side of the base.
The polygon having the least number of sides is a triangle, which has 3 sides. So, the prism having the least number of faces is a triangular prism.
Number of faces = 2 + 3 = 5
Hence, option 1 is the correct option.
3
Without counting, any faces, edges or vertices, tell which of the following objects will not satisfy Euler's formula:
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Euler's formula holds for a polyhedron, that is, for a solid which is made up of polygonal regions.
The solids given in options 1, 3 and 4 are an octagonal prism, an octahedron and a square pyramid respectively. Each of them is made up of polygonal regions, so each of them is a polyhedron.
The solid given in option 2 is a cone. It has a circular base and a curved surface, so it is not made up of polygonal regions and hence it is not a polyhedron.
So, the cone will not satisfy Euler's formula.
Hence, option 2 is the correct option.
4
The given polygon is the base of a pyramid. The number of faces this pyramid will have:
- (a)5
- (b)6
- (c)8
- (d)10
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The given polygon is a pentagon, which has 5 sides.
So, the pyramid having it as its base is a pentagonal pyramid.
Number of faces in a pyramid is 1 more than the number of sides of the polygonal base.
Number of faces = 5 + 1 = 6
Hence, option 2 is the correct option.
5
The sum of the interior angles of a polygon which is the base of a prism is 540°. The prism will have:
- (a)10 vertices
- (b)8 vertices
- (c)7 vertices
- (d)6 vertices
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Let the number of sides of the polygon forming the base of the prism be $n$.
Sum of the interior angles of a polygon of $n$ sides $= (n - 2) \times 180^\circ$
According to the question,
$$\begin{aligned}\Rightarrow (n - 2) \times 180^{\circ} = 540^{\circ} \\[1em] \Rightarrow n - 2 = \frac{540^{\circ}}{180^{\circ}}\\[1em] \Rightarrow n - 2 = 3 \\[1em] \Rightarrow n = 3 + 2 \\[1em] \Rightarrow n = 5\end{aligned}$$
So, the base of the prism is a pentagon and the prism is a pentagonal prism.
Number of vertices in a prism is twice the number of sides of the polygonal base.
Number of vertices = 5 × 2 = 10
Hence, option 1 is the correct option.
6
Archana, a class 8 student, claims that we cannot have a polyhedron with 8 faces, 12 vertices and 18 edges. Her statement is:
- (a)correct
- (b)incorrect
- (c)cannot say anything
- (d)none of these
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Given:
F = 8, E = 18, V = 12
According to Euler's relation,
F - E + V = 2
L.H.S. = F - E + V = 8 - 18 + 12 = 2
R.H.S. = 2
Since, L.H.S. = R.H.S.
So, Euler's relation is satisfied and hence we can have a polyhedron with 8 faces, 12 vertices and 18 edges.
So, the claim made by Archana is incorrect.
Hence, option 2 is the correct option.
7
When you cut a corner of a cube as shown, what will be the shape of the cutout piece?
- (a)Triangular prism
- (b)Triangular pyramid
- (c)Octahedron
- (d)none of these
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At a corner of a cube, 3 faces of the cube meet. The cut is made along a plane which meets the 3 edges of the cube passing through that corner.
So, the cutout piece is bounded by 3 triangular faces, which are parts of the 3 faces of the cube meeting at that corner, and 1 more triangular face, which is formed by the cut.
So, the cutout piece has 4 triangular faces, 6 edges and 4 vertices.
So, the cutout piece is a triangular pyramid.
Hence, option 2 is the correct option.
8
Identical cubes are stacked in the corner of a room as shown.
How many cubes are not visible?
- (a)10
- (b)11
- (c)15
- (d)20
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From the figure, the cubes are stacked in 5 layers, one above the other.
In the bottom layer, the cubes are placed in 5 rows having 5, 4, 3, 2 and 1 cubes.
Number of cubes in the bottom layer = 5 + 4 + 3 + 2 + 1 = 15
Number of cubes in the second layer = 4 + 3 + 2 + 1 = 10
Number of cubes in the third layer = 3 + 2 + 1 = 6
Number of cubes in the fourth layer = 2 + 1 = 3
Number of cubes in the top layer = 1
Total number of cubes = 15 + 10 + 6 + 3 + 1 = 35
In each layer, only the cubes lying along the front edge of that layer can be seen, as every other cube is covered by the cubes lying in front of it and by the cube lying above it.
Number of cubes which are visible = 5 + 4 + 3 + 2 + 1 = 15
Number of cubes which are not visible = 35 - 15 = 20
Hence, option 4 is the correct option.
9
The number of faces in the given shape is:
- (a)6
- (b)8
- (c)9
- (d)10
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The given shape is obtained by cutting a rectangular groove along the length of a cuboid.
So, its cross-section is a rectangle from which a rectangle has been cut out at the top, and this cross-section has 8 sides.
The given shape has 1 face along each side of this cross-section.
Number of such faces = 8
Besides these, the given shape has 2 more faces, one at each of its two ends.
Number of faces = 8 + 2 = 10
Hence, option 4 is the correct option.
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