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# Volume of Cones – Explanation & Examples

**In geometry, a cone is a 3-dimensional shape with a circular base and a curved surface that tapers from the base to the apex or vertex at the top. In simple words, a cone is a pyramid with a circular base.**

Common examples of cones are ice-cream cones, traffic cones, funnels, tipi, castle turrets, temple tops, pencil tips, megaphones, Christmas trees, etc.

In this article, we will discuss how to use the volume of a cone formula to calculate the volume of a cone.

## How to Find the Volume of a Cone?

In a cone, the perpendicular length between the vertex of a cone and the center of the circular base is known as the** height** (**h**) **of a cone**. A cone’s slanted lines are the **length** (**L**) **of a cone** along the taper curved surface. All of these parameters are mentioned in the figure above.

T**o find the volume of a cone, you need the following parameters:**

**Radius**(**r**) of the circular base,- The height or the slanted height of a cone.

Like all other volumes, the volume of a cone is also expressed in cubic units.

### Volume of a cone formula

The volume of a cone is equal to one-third of the base area’s product and the height. The formula for the volume is represented as:

**Volume of a cone = ****⅓ x πr ^{2} x h**

**V = ⅓** **πr ^{2} h**

Where V is the volume, r is the radius and h, is the height.

The slant height, radius, and height of a cone are related as;

**Slant height of a cone, L = √(r ^{2}+h^{2}) **………. (Pythagorean theorem)

Let’s gain an insight into the volume of a cone formula by working out a few example problems.

*Example 1*

Find the volume of the cone of radius, 5 cm, and height, 10 cm.

__Solution__

By the volume of a cone formula, we have,

⇒V = ⅓ πr^{2}h

⇒V = ⅓ x 3.14 x 5 x 5 x 10

= 262 cm^{3}

*Example 2*

The radius and slant height of a cone are 12 mm and 25 mm. respectively. Find the volume of the cone.

__Solution__

Given:

Slant height, L= 25 mm

radius, r = 12 mm

L = √ (r^{2 }+ h^{2})

By substitution, we get,

⇒25 = √ (12^{2 }+ h^{2})

⇒25 = √ (144 + h^{2})

Square both sides

⇒625 = 144 + h^{2}

Subtract by 144 on both sides.

481 = h^{2}

√481 = h

h = 21.9

Hence, the height of the cone is 21.9 mm.

Now, calculate the volume.

Volume = ⅓ πr^{2}h

= ⅓ x 3.14 x 12 x 12 x 21.9

= 3300.8 mm^{3}.

*Example 3*

A conical silo of radius 9 feet and height 14 feet releases cereals at its bottom at a constant rate of 20 cubic feet per minute. How long will it take for the silo to be empty?

__Solution__

First, find the volume of the conical silo

Volume = ⅓ x 3.14 x 9 x 9 x 14

= 1186.92 cubic feet.

To get the time take for the silo to be empty, divide the volume of the silo by the flow rate of the cereals.

= 1186.92 cubic feet/20 cubic feet per minute

= 59 minutes

*Example 4*

A conical storage tank has a diameter of 5 m and a height of 10 m. Find the capacity of the tank in liters.

__Solution__

Given, diameter = 5 m ⇒ radius = 2.5 m

Height = 10 m

Volume of a cone = ⅓ πr^{2}h

= ⅓ x 3.14 x 2.5 x 2.5 x 10

= 65.4 m^{3}

Since, 1000 liters = 1 m^{3}, then

65.4 m^{3} = 65.4 x 1000 liters

= 65400 liters.

*Example 5*

A solid plastic sphere of radius 14 cm is melted down into a cone of height, 10 cm. What will be the radius of the cone?

__Solution__

Volume of the sphere = 4/3 πr^{3}

= 4/3 x 3.14 x 14 x 14 x 14

= 11488.2 cm^{3}

The cone will also have the same volume of 11488.2 cm^{3}

Therefore,

⅓ πr^{2}h = 11488.2 cm^{3}

⅓ x 3.14 x r^{2} x 10 = 11488.2 cm^{3}

10.5r^{2 }= 11488.2 cm^{3}

r^{2} = 1094

r = √1094

r = 33

Therefore, the radius of the cone will be 33 cm.

*Example 6*

Find the volume of the cone, whose radius is 6 feet and height is 15 feet

__Solution__

Volume of a cone = 1/3 x 3.14 x 6 x 6 x 15

= 565.2 ft^{3}.