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.
Related Topics: More Geometry Lessons

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.

sphere

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:

vol & surface area of sphere

where r is the radius of the sphere.

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

Example:

Calculate the volume of sphere with radius 4 cm.

Solution:

Volume of sphere

volume 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.



A ball has a diameter of 18 cm. a) Sketch a cylinder that fits the ball and label its height and base. b) What is the volume of the cylinder? c) What is the volume of the ball?



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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