# Cone

Cone is an object which is hollow, but tapers down to a point at the other end known as the vertex.

A Cone is generated by rotating a right triangle in a clock wise direction or anticlockwise direction.

## Right circular cone

A Right circular cone is again generated by rotating a right angled triangle rotating clockwise or a clockwise.

## Surface Area of a cone

Area of cone has the following specifications

Surface Area of cone = π b l where l2 = b2 + h2

## Cone formula

There are three formulas related to cone

1. Surface area of cone = π b l where b is the radius and l is the slant height.
2. Volume of cone = $\frac{1}{3}$ π b2 h. where h is the height of the cone.
3. Base area of cone = π b2.

## Volume of a cone

Volume of cone = $\frac{1}{3}$ π b2 h. where h is the height of the cone.

## Problems on volume of a cone

1. Find the surface Area of the cone whose base radius is 3cm and height is 4 cm.

Answer: Surface area of cone is given by π b l where b is base radius and l is the slant height.
l2 = b2 + h2
= 32 + 42
= 9+16
= 25
l = √25
l=5 cm
Surface Area of the cone = π b l = $\frac{22}{7}$ $\times 3 \times 5$
= 47.14 sq.cm.

2. Find the surface Area of the cone whose base radius is 5cm and height is 12cm.

Answer: Surface area of cone is given by π b l where b is base radius and l is the slant height.
l2 = b2 + h2
= 52 + 122
= 25+144
= 169
l = √169
l=13cm
Surface Area of the cone = π b l = $\frac{22}{7}$ $\times 5 \times 13$
= 204.28 sqcm.

3. Find the volume of the cone whose base radius is 6 cm and height is 12 cm.

Answer: The volume of the cone is given by $\frac{1}{3}$ π b2 h.
$\frac{1}{3}$ $\times$ $\frac{22}{7}$ $\times$ 62 $\times$ 12 = $\frac{9504}{21}$
= 452.57 cubic cm.

4. Find the volume of the cone whose base radius is 2 cm and height is 21 cm.

Answer: The volume of the cone is given by $\frac{1}{3}$ π b2 h.
$\frac{1}{3}$ $\times$ $\frac{22}{7}$ $\times$ 22 $\times$ 21 = $\frac{1848}{21}$
= 616 cubic cm

5. Find the volume of the cone whose base radius is 4cm and height is 35 cm.

Answer: The volume of the cone is given by $\frac{1}{3}$ π b2 h.
$\frac{1}{3}$ $\times$ $\frac{22}{7}$ $\times$ 42 $\times$ 35 = $\frac{1760}{7}$
= 251.42cubic cm.

6. Find the surface Area of the cone whose base radius is 5cm and height is 6cm.

Answer: Surface area of cone is given by π b l where b is base radius and l is the slant height.
l2 = b2 + h2
= 52 + 62
=25+36
= 71
l = √71
l=8.42cm
Surface Area of the cone = π b l = $\frac{22}{7}$ $\times 3 \times$ 8.42
= 79.44 square cm.

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