**Visualising Solid Shapes**

**Question 1.****Can a polyhedron have for its faces(i)** 3 triangles?

**(ii)**4 triangles?

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

**Solution:**

Since a polyhedron is a solid, which is bounded by four or more polygonal faces in such a way that pairs of faces meet along edges and three or more edges meet in each vertex, therefore,**(i)** A polyhedron cannot for its faces have three triangles.**(ii)** A polyhedron can for its faces have four triangles.**(iii)** A polyhedron can for its faces have a square and four triangles.

**Question 2.****Is it possible to have a polyhedron with any given number of faces? (Hint : Think of a pyramid)**

**Solution:**

Yes, it is possible only if the number of faces are greater than or equal to four.

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

**Solution:**

We know that a prism is a polyhedron, two of whose faces are congruent polygons in parallel planes and whose other faces are parallelograms. Therefore,

**(i)**A nail is not a prism.

**(ii)**An unsharpened pencil is a prism.

**(iii)**A table weight is not a prism.

**(iv)**A box is a prism.

**Question 4.****(i)** How are prisms and cylinders alike?**(ii)** How are pyramids and cones alike?

**Solution:****(i)** A prism becomes a cylinder provided the number of sides of its base becomes larger and larger.**(ii)** A pyramid becomes a cone provided the number of sides of its base becomes larger and larger.

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

**Solution:**

Yes, it can be a cube. But it can be a cuboid also.

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

**Solution:**

**(i) In this figure,**

F = 7, V = 10, E =15

∴ F + V =7 + 10 = 17 and E + 2 =15 + 2 = 17

⇒ F+V=E+ 2

Hence, Euler’s formula is verified.

**(ii) In this figure,**

F =9, V = 9,E =16

∴ F + V =9 + 9 =18 and E + 2 =16 +2 =18

⇒ F + V =E + 2

Hence, Euler’s formula is verified.

**Question 7.****Using Euler’s formula find the unknown.****Solution:****In 1st part :** Faces = ?, Vertices =6, Edges =12

By Euler’s formula, we know that F + V= E + 2

⇒ F = E + 2 – V

= 12 + 2-6 =8

Hence, faces =8

**In 2nd part :** Faces = 5, Vertices = ?, Edges = 9

By Euler’s formula, we know that

F + V = E + 2

⇒ V = E + 2 – F = 9+ 2- 5 = 6

Hence, vertices =6

**In 3rd part :** Faces =20, Vertices =12, Edges =?

By Euler’s formula, we know that

F+V=E+2

⇒ E = F + V – 2 = 20 + 12 – 2 = 30

Hence, edges =30

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

**Solution:**

Since F + V ≠ E +2 (as 10 +15 ≠ 20 + 2)

A polyhedron cannot have 10 faces, 20 edges and 15 vertices.