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# Volume of Spheres – Explanation & Examples

**The sphere is an extended version of a circle. Or it will be right to say a 3D version of a circle. In geometry, a sphere is a 3-dimensional round solid figure in which every point on its surface is equidistant from its center. **

Common examples of objects which spherical in shape include balls, globes, ball bearings, water drops, bubbles, planets, etc.

In this article, we discuss how to find the volume of a sphere using the volume of a sphere formula.

## How to Find the Volume of a Sphere?

**The volume of a sphere is the amount of space occupied by it.** For a hollow sphere like a football, the volume can be viewed as the number of cubic units required to fill up the sphere.

**To find the volume of a sphere, you only need to know the radius of the sphere**.

**The volume of a sphere is measured in cubic units, i.e., m ^{3}, cm^{3}, in^{3}, ft^{3,} etc.**

### Volume of a sphere formula

The volume of a sphere formula is given as:

**Volume of a sphere = 4/3 πr ^{3}**

where, π = 3.14 and r = radius of a sphere.

A half of a sphere is known as a hemisphere. The volume of a hemisphere is equal to half the volume of a sphere i.e.

**Volume of a hemisphere = ½ (4/3) πr ^{3}**

**= 2/3 πr ^{3}**

The volume of a sphere formula is attributed to Archimedes Principle, which states that:

When a solid object is completely submerged in a container filled with water, the displaced water volume is equal to the volume of the spherical solid object.

Let’s gain an insight into the volume of a sphere formula by solving a couple of example problems.

*Example 1*

Find the volume of a sphere whose radius is 5 cm.

__Solution__

By, the volume of a sphere formula, we have

V = 4/3 πr^{3}

= (4/3) x 3.14 x 5^{3}

= (4/3) x 3.14 x 5 x 5 x 5

= 523.3 cm^{3}

*Example 2*

What is the volume of a sphere with a radius of 24 mm?

__Solution__

Since we know the radius is half of the diameter, then

r = 24/2 = 12 mm

Volume of a sphere = 4/3 πr^{3}

By substitution, we get

V = (4/3) x 3.14 x 12 x 12 x 12

= 7734.6 mm^{3}

*Example 3*

The volume of a sphere is 523 cubic yards. Find the radius of the sphere.

__Solution__

Given, V = 523 cubic yards

Volume of a sphere, V = 4/3 πr^{3}

523 = (4/3) x 3.14 x r^{3}

523 = 4.19r^{3}

Divide both sides by 4.19

r^{3} = 124.82

^{3}√r^{3} = ^{3}√124.82

r = 5

So, the radius of the sphere is 5 yards.

*Example 4*

A spherical solid metal of a radius of 8 cm is melted down into a cube. What will be the dimensions of the cube?

__Solution__

Equate the volume the sphere to the volume of the cube

4/3 πr^{3} = a^{3}

4/3 x 3.14 x 8 x 8 x 8 = a^{3}

2143.6 = a^{3}

^{3}√2143.6 =^{3}√a^{3}

a =12.9

Therefore, the sides of the cube will be 12.9 cm.

*Example 5*

The radius of an inflated spherical balloon is 7 feet. Suppose air is leaking from the balloon at a constant rate of 26 cubic feet per minute. How long will it take for the balloon to be completely deflated?

__Solution__

Volume of the spherical balloon = 4/3 πr^{3}

= 4/3 x 3.14 x 7 x 7 x 7

= 1436.03 cubic feet

Divide the volume of the balloon by the rate of leakage

Time in minutes = 1436.03 cubic feet/26 cubic feet

= 55 minutes

*Example 6*

What will be the radius of a sphere with the same volume as a rectangular prism of length 5 mm, width, 3 mm, and height 4 mm?

__Solution__

Equate the volume of the rectangular prism to the volume of the sphere.

Volume of the prism = 5 x 3 x 4

= 60 mm^{3}

Therefore,

60 = 4/3 πr^{3}

60 = 4/3 x 3.14 x r^{3}

60 = 4.19r^{3}

r^{3 }= 14.33

r = ^{3}√14.33

r = 2.43

Hence, the radius of the sphere will be 2.43 mm.

*Example 7*

The water level in a cylindrical container of radius 0.5 m is 3.2 m. When a spherical solid object is completely submerged in the water, the water level rises by 0.6 m. Find the volume of the sphere.

__Solution__

The volume of the displaced water = volume of the sphere.

Volume of the displaced water in the cylinder = πr^{2}h

= 3.14 x 0.5 x 0.5 x 0.6

= 0.471 m^{3}.

*Example 8*

The volume of a typical baseball is 230 cm^{3}. Find the radius of the ball.

__Solution__

Volume of a sphere = 4/3 πr^{3}

230 = 4/3 x 3.14 x r^{3}

230 = 4.19r^{3}

r^{3} = 54.9

r = ^{3}√54.9

r = 3.8

Thus, the radius of the baseball is 3.8 cm

*Example 9*

Find the volume of a hemisphere whose diameter is 14 in.

__Solution__

Volume of a hemisphere = 2/3 πr^{3}

V = 2/3 x 3.14 x 7 x 7 x 7

= 718 cubic inches