Volume of a Sphere

Solid geometry is concerned with three-dimensional shapes. In this lesson, we will learn

• how to find the volume of a sphere.
• how to find the volume of a hemisphere.
• how to prove the formula for the volume of a sphere.

Spheres

A sphere is a solid with all its points the same distance from the center. The distance is known as the radius of the sphere. The maximum straight distance through the center of a sphere is known as the diameter of the sphere. The diameter is twice the radius.

The volume of a sphere is equal to four-thirds of the product of pi and the cube of the radius.

The volume and surface area of a sphere is given by the formulas:

where r is the radius of the sphere.

Worksheet to calculate the volume of spheres.
Worksheet to calculate the surface area of spheres.

Example:

Calculate the volume of sphere with radius 4 cm.

Solution:

Volume of sphere

We can also change the subject of the formula to obtain the radius given the volume.

Example:

The volume of a spherical ball is 5,000 cm³. What is the radius of the ball?

Solution:

The following videos explain how to use the formula to calculate the volume of a sphere.

Volume of a hemisphere

A hemisphere is half a sphere, with one flat circular face and one bowl-shaped face.

Volume of hemisphere

where r is the radius

This video shows how to find the volume of a hemisphere.

Word Problems about Spheres

The following video shows how to solve problems involving the formulas for the area and volume of spheres.

A sphere has a volume of 288π. Find its area.

This video gives a review of the formula for determining the volume of a sphere accompanied by practice problems.

Proof for the Formula of the Volume of a Sphere

This video gives a proof for the formula of the volume of a sphere that does not involve calculus. This is an "approximated" proof. You would need to use calculus for a more rigorous proof.

This video derives the formula of a sphere using calculus.

This video shows how Archimedes derived the volume of a sphere. To do so, he had to use a formula for the volume of a cone.

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