**Question 1: What is the least number of planes that can enclose a solid? What is the name of the solid?**

**Solution:**

The least number of planes that are required to enclose a solid is 4.

The name of the solid is tetrahedron. It is a solid with four planes.

**Question 2: Can a polyhedron have for its faces?**

**(i) 3 triangles?**

**Solution:**

No, because we need minimum 4 triangular faces in order to complete a polyhedron.

**(ii) 4 triangles?**

**Solution:**

Yes, A tetrahedron has 4 triangles as its faces.

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

**Solution:**

Yes, A square pyramid has a square and four triangles as its faces.

**Question 3: Is it possible to have a polyhedron with any given number of faces?**

**Solution:**

Yes, if the number of faces is equal to or more than 4. As there is no possible polyhedron with 3 or less faces.

**Question 4: Is a square prism same as a cube?**

**Solution:**

Yes, a square prism is same as a cube. Both of them have 6 faces, 8 vertices and 12 edge.

The only difference is that a square prism is a 3-d shape with six rectangular shaped sides, out of which two are squares and

a cube is a rectangular prism having same length, width and height.

**Question 5: Can a polyhedron have 10 faces, 20 edges and 15 vertices?**

**Solution:**

No.

Reason –

Given,

Number of faces (F) = 10

Number of edges (E) = 20

Number of vertices (V) = 15

We know that, every polyhedron satisfies Euler’s formuala.

So, By using Euler’s formula –

V + F = E + 2

15 + 10 = 20 + 2

25 ≠ 22

Since, the given polyhedron does not satisfy Euler’s formula. Thus, a polyhedron can not have 10 faces, 20 edges and 15 vertices.

**Question 6: Verify Euler’s formula for each of the following polyhedrons:**

**(i)**

**Solution:**

In the given polyhedron –

Number of faces (F) = 7

Number of edges (E) = 15

Number of vertices (V) = 10

Now, By using Euler’s formula –

V + F = E + 2

10 + 7 = 15 + 2

17 = 17

Here, Euler’s formula is satisfied.

Hence, verified.

**(ii)**

**Solution:**

In the given polyhedron –

Number of faces (F) = 9

Number of edges (E) = 16

Number of vertices (V) = 9

Now, By using Euler’s formula –

V + F = E + 2

9 + 9 = 16 + 2

18 = 18

Here, Euler’s formula is satisfied.

Hence, verified.

**(iii)**

**Solution:**

In the given polyhedron –

Number of faces (F) = 9

Number of edges (E) = 21

Number of vertices (V) = 14

Now, By using Euler’s formula –

V + F = E + 2

14 + 9 = 21 + 2

23 = 23

Here, Euler’s formula is satisfied.

Hence, verified.

**(iv)**

**Solution:**

In the given polyhedron –

Number of faces (F) = 8

Number of edges (E) = 12

Number of vertices (V) = 6

Now, By using Euler’s formula –

V + F = E + 2

6 + 8 = 12 + 2

14 = 14

Here, Euler’s formula is satisfied.

Hence, verified.

**(v)**

**Solution:**

In the given polyhedron –

Number of faces (F) = 9

Number of edges (E) = 16

Number of vertices (V) = 9

Now, By using Euler’s formula –

V + F = E + 2

9 + 9 = 16 + 2

18 = 18

Here, Euler’s formula is satisfied.

Hence, verified.

**Question 7: Using Euler’s formula find the unknown:**

Faces | ? | 5 | 20 |

Vertices | 6 | ? | 12 |

Edges | 12 | 9 | ? |

**Solution:**

(i)Given,

Number of vertices (V) = 6

Number of edges (E) = 12

By using Euler’s formula –

V + F = E + 2

6 + F = 12 + 2

F = 14 – 6

F = 8

Thus, the number of faces is 8.

(ii)Given,

Number of faces (F) = 5

Number of edges (E) = 9

By using Euler’s formula –

V + F = E + 2

V + 5 = 9 + 2

V = 11 – 5

V = 6

Thus, the number of vertices is 6.

(iii)Given,

Number of vertices (V) = 12

Number of faces (F) = 20

By using Euler’s formula –

V + F = E + 2

12 + 20 = E + 2

E = 32 – 2

E = 30

Thus, the number of edges is 30.