Math Core

Basketballs, marbles, bubbles and planets are all shaped like spheres. A sphere has no flat base, so "base times height" can't work directly. But a sphere fits neatly inside a cylinder, and that connection gives you its volume.

What is a sphere?

A sphere is the set of all points in space that are the same distance from a center point. That distance is the radius, $r$ . A segment through the center from one side to the other is a diameter, and it is twice the radius.

A sphere with radius r. The oval around the middle shows where a flat cut through the center would go.

Fitting a sphere in a cylinder

Place a ball snugly inside a can so that it touches the sides, the bottom and the lid. The can is a cylinder with:

radius $r$ (the same as the ball), and
height $2r$ (the ball's diameter).

The can's volume is $\pi r^2 \cdot 2r = 2\pi r^3$ . More than 2,000 years ago, the Greek mathematician Archimedes discovered that the sphere fills exactly two thirds of this cylinder:

V_{\text{sphere}} = \frac{2}{3} \cdot 2\pi r^3 = \frac{4}{3}\pi r^3

Here is a way to remember it. A cone with the same radius and height $2r$ fills one third of the can. The sphere fills two thirds. Together, the cone and the sphere fill the can exactly.

Volume of a sphere

V = \frac{4}{3}\pi r^3

where $r$ is the radius. The radius is cubed, because a sphere grows in all three directions.

Worked example: From the radius

Find the volume of a sphere with a radius of 3 cm, in terms of $\pi$ and to the nearest tenth.

\begin{aligned} V &= \tfrac{4}{3}\pi r^3 \\ &= \tfrac{4}{3} \cdot \pi \cdot 3^3 \\ &= \tfrac{4}{3} \cdot 27\pi \\ &= 36\pi \approx 113.1 \end{aligned}

The volume is $36\pi$ , or about $113.1$ cubic centimeters.

Worked example: From the diameter

A ball has a diameter of 12 inches. Find its volume in terms of $\pi$ .

The radius is $12 \div 2 = 6$ inches, and $6^3 = 216$ .

V = \tfrac{4}{3} \cdot 216\pi = 288\pi

The ball holds $288\pi$ cubic inches of air.

Common mistake

$r^3$ means $r \cdot r \cdot r$ , not $3r$ . For $r = 6$ , $r^3 = 216$ , not 18. And as always, check whether you were given the radius or the diameter before you cube.

Hemispheres

A hemisphere is half of a sphere, like a bowl or a dome. Its volume is half of the sphere's:

V_{\text{hemisphere}} = \frac{1}{2} \cdot \frac{4}{3}\pi r^3 = \frac{2}{3}\pi r^3

Worked example: A dome

A dome is a hemisphere with a radius of 9 m. Find the volume of air inside, in terms of $\pi$ .

$9^3 = 729$ , so

V = \tfrac{2}{3} \cdot 729\pi = 486\pi

The dome holds $486\pi$ cubic meters of air.

Combining solids

Many objects are made of pieces you already know. Find each volume and add.

Worked example: An ice cream cone

An ice cream cone is 8 cm tall with a radius of 3 cm. A hemisphere of ice cream with the same radius sits on top. Find the total volume in terms of $\pi$ .

\begin{aligned} \text{Cone: } & \tfrac{1}{3} \cdot \pi \cdot 3^2 \cdot 8 = 24\pi \\ \text{Hemisphere: } & \tfrac{2}{3} \cdot \pi \cdot 3^3 = 18\pi \\ \text{Total: } & 24\pi + 18\pi = 42\pi \end{aligned}

The total volume is $42\pi$ cubic centimeters.

Working backward

If you know a sphere's volume, you can find its radius. Solve for $r^3$ , then take the cube root. For example, if $\tfrac{4}{3}\pi r^3 = 36\pi$ , multiply both sides by $\tfrac{3}{4}$ to get $\pi r^3 = 27\pi$ , so $r^3 = 27$ and $r = \sqrt[3]{27} = 3$ .

Tip

Doubling the radius of a sphere multiplies its volume by $2^3 = 8$ , not by 2. A ball twice as wide holds eight times as much.

Practice

Practice 1

Find the volume of a sphere with a radius of 9 ft, in cubic feet. Give your answer in terms of $\pi$ .