Lesson 3: Volumes of Spheres

Property

A sphere of radius $r$ is the set of all points in space that are of a distance $r$ from a point $C$, called the center of the sphere. The volume of a sphere of radius $r$ is $V = \frac{4}{3} \pi r^3$.

Examples

• A basketball has a radius of 12 cm. Its volume is $V = \frac{4}{3} \pi (12^3) = \frac{4}{3} \pi (1728) = 2304\pi$ cubic cm.
• A hemisphere is half of a sphere. If a bowl in the shape of a hemisphere has a radius of 5 inches, its volume is $V = \frac{1}{2} \cdot \frac{4}{3} \pi (5^3) = \frac{2}{3} \pi (125) = \frac{250}{3}\pi$ cubic inches.
• A spherical balloon has a volume of $36\pi$ cubic feet. To find its radius, we solve $36\pi = \frac{4}{3} \pi r^3$. This simplifies to $27 = r^3$, so the radius is 3 feet.

Explanation

This formula calculates the space inside a perfectly round ball. The only measurement you need is the radius, which is the distance from the center to the surface. Notice the radius is cubed ($r^3$), reflecting its three-dimensional nature.

Property

The volume of a sphere is $\frac{2}{3}$ times the volume of a cylinder that has the same diameter as the sphere and height equal to the diameter:

When $h = 2r$ (diameter), this becomes:

Property

To find the radius of a sphere when given its volume, solve the equation $V = \frac{4}{3}\pi r^3$ for $r$: