Table of Contents

Toggle## Class 8, Maths, Chapter 10, Exercise 10.3 Solutions

**Q.1. Can a polyhedron ****have for its faces**

**(i) 3 triangles? **

**(ii) 4 triangles?**

**(iii) a square and four triangles?**

**Ans: **(i) No, because such a polyhedron is not possible. A polyhedron has minimum 4 faces.

(ii) Yes, because a triangular pyramid has 4 triangular faces.

(iii) Yes, because a square pyramid has a square face and 4 triangular faces.

**Q.2. Is it possible to have a polyhedron with any given number of faces?**

**Ans: **yes, only if the number of faces are greater or equal to four. for example: Pyramid which has four faces.

**Q. 3. Which are prisms among the following?**

**Ans: **

**Only (ii) unsharpened pencil and (iv) a box are the prism.**

**How:**

(i) No, A nail cannot be a prism because there is only curved side.

(ii) Yes, An unsharped pencil is a prism because It has two identical hexagonal base and flat sides.

(iii) No, A table weight is not a prism because it doesn’t have two identical bases.

(iv) Yes, A box is a prism because It has two identical hexagonal base and flat sides.

**Explanation:**

**Let us know about the Prism: **

**Prism**: it is a solid object with two faces (called the bases) that are congruent and parallel and where all cross- section are also congruent to the bases.

**Q.4. (i) How are prisms and cylinders alike?**

**(ii) How are pyramids and cones alike?**

**Ans: **(i) A prism becomes cylinder as the number of sides are increased to certain extent.

(ii) A pyramid becomes a cone as the number of sides are increased to certain extent.

**Q.5. Is a square prism same as a cube? Explain.**

**Ans: **No, a square prism and a cube are not the same.

A square prism is a shape with six rectangular faces, where three of the faces are congruent squares, and the other three faces are rectangles with the same height and width.

A cube, on the other hand, a cube is a square prism with equal length, width, and height.

So, a cube is a specific type of square prism, but not all square prisms are cubes.

**Q.6. Verify Euler’s formula for these solids.**

**Ans: ****(i) **Faces = 7, Sides = 15, Vertices = 10

**Euler’s formula: F + V – E = 2**

⇒ 7 + 10 – 15 = 2

⇒ 2 = 2

Hence, Euler’s formula is verified.

**(ii) **Faces = 9, Sides = 16, Vertices = 9

**Euler’s Formula: F + V – E = 2**

⇒ 9 + 9 – 16 = 2

⇒ 2 = 2

Hence, Euler’s formula is verified.

**Q.7. Using Euler’s formula find the unknown.**

**Ans: **Using Eulers Formula: F + V – E = 2

**(i) F + 6 – 12 = 2**

F – 6 = 2

F = 8

**(ii) 5 + V – 9 = 2**

V – 4 = 2

V = 6

**(iii) 20 + 12 – E = 2**

32 – E = 2

E = 30

Thus, the table can be completed as:

**Q.8. Can a polyhedron have 10 faces, 20 edges and 15 vertices?**

**Ans: **Here, Number of faces, F = 10,

Number of Edges, E = 20,

Number of Vertices, V = 15

According to Euler’s Formula:

F + V – E = 2

⇒ 10 + 15 – 20

= 25 – 20

⇒ 5 ≠ 2

Since Euler’s formula is not satisfied, such a polyhedron is not possible. Which means a polyhedron do not have 10 Faces, 20 Edges and 15 Vertices.