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**Question 1:** Which of the following solids are not polyhedrons?

**Solution: **

(i) A cylinder is not a polyhedron.

(ii) A cuboid is a polyhedron.

(iii) A cube is a polyhedron.

(iv) A cone is not a polyhedron.

(v) A sphere is not a polyhedron.

(vi) A pyramid is a polyhedron.**Question 2: **Why the following solids are not polyhedron?

(i) A sphere.

(ii) A cone.

(iii) A cylinder.**Solution:** Since, a polyhedron is a solid shape bounded by polygons. However,

(i) a sphere,

(ii) a cone and

(iii) a cylinder are not polyhedron because they are made of polygons, i.e. their faces are not polygons.**Question 3: **Name the following polyhedron.

How many faces, vertiecs and edges of this solid are there?**Solution:** ∵ The ends (bases) of the given solid are congruent rectilinear figure each of six sides.

∴ It is a hexagonal prism.

In a hexagonal prism, we have:

The number of faces = 8

The number of edges = 18

The number of vertices = 12**Question 4: **What Euler’s formula? Verify the Euler’s formula for a pentagonal prism.**Solution:** If a polyhedron is having number of faces as F, number of edges as E and the number of vertices as V, then the relationship

F + V = E + 2

is known as Euler’s formula. Following figure is a solid pentagonal prism

It has:

Number of faces (F) = 7

Number of edges (E) = 15

Number of vertices (V) = 10

Substituting the values of F, E and V in the relation,

F + V = E + 2

we have

7 + 10 = 15 + 2

⇒ 17 = 17

Which is true, the Euler’s formula is verified.**Question 5: **A polyhedron is having 8 vertices and 12 edges. How many faces of it are there?**Solution: **Number of vertices (V) = 8

Number of edges (E) = 12

Let the number of faces = F

Now, using Euler’s formula

F + V = E + 2

we have

F + 8 = 12 + 2

⇒ F + 8 = 14

⇒ F = 14 – 8 ⇒ F = 6

Thus, the required number of faces = 6.**Question 6: **An icosahedron is having 20 triangular faces and 12 vertices. Find the number of its edges.**Solution:** Here:

Number of faces (F) = 20

Number of vertices (V) = 12

Let the number of edges be E.

∴ Using Euler’s formula, we have

F + V = E + 2

⇒ 20 + 12 = E + 2

⇒ 32 = E + 2

⇒ E = 32 – 2 = 30

Thus, the required number of edges = 30.**Question 7: **What is the least number of planes that can enclose a solid? Name the simplest regular polyhedron and verify Euler’s formula for it.**Solution:** At least 4 planes can form to enclose a solid. Tetrahedron is the simple polyhedron. Following figure represents a simplest solid, called tetrahedron.

A tetrahedron has:

4 triangular faces, i.e. F = 4

4 vertices, i.e. V = 4

6 edges, i.e. E = 6

Now, substituting the values of F, V and E in Euler’s formula, i.e.

F + V = E + 2

we have

4 + 4 = 6 + 2

⇒ 8 = 8, which is true.

Thus, Euler’s formula is verified for a tetrahedron.

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