Math Core

Balls, bubbles, planets and ball bearings are all close to spheres. A sphere has no flat base, so $V = Bh$ can't be used directly. Instead, a clever comparison with a cylinder and a cone, using Cavalieri's principle, pins down its volume exactly.

The parts of a sphere

A sphere is the set of all points in space at a distance $r$ from a fixed point, its center. The distance $r$ is the radius, and a segment through the center with endpoints on the sphere is a diameter, of length $2r$ .

A sphere with radius r. Every point on the surface is r units from the center.

A plane through the center cuts the sphere in a great circle, the largest circle that can be drawn on it. The great circle splits the sphere into two congruent hemispheres.

Where the formula comes from

Take a hemisphere of radius $r$ resting on its flat face. Next to it, place a cylinder with radius $r$ and height $r$ , and carve out of the cylinder a cone whose apex touches the center of the cylinder's bottom and whose base is the cylinder's top.

Now slice both solids with a horizontal plane at height $y$ above the table.

Hemisphere: the slice is a circle. By the Pythagorean theorem, its radius is $\sqrt{r^2 - y^2}$ , so its area is $\pi(r^2 - y^2)$ .
Cylinder minus cone: the slice is a ring. The outer circle has radius $r$ . The cone widens at the same rate it rises, so at height $y$ its radius is $y$ . The ring's area is $\pi r^2 - \pi y^2 = \pi(r^2 - y^2)$ .

The areas match at every height. By Cavalieri's principle, the two solids have the same volume:

V_{\text{hemisphere}} = \pi r^2 \cdot r - \tfrac{1}{3}\pi r^2 \cdot r = \pi r^3 - \tfrac{1}{3}\pi r^3 = \tfrac{2}{3}\pi r^3

A whole sphere is two hemispheres, so its volume is $\tfrac{4}{3}\pi r^3$ .

Volume of a sphere

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

The derivation also shows a surprising fact that Archimedes was proud of: a sphere fills exactly $\tfrac{2}{3}$ of the smallest cylinder that holds it. That cylinder has radius $r$ and height $2r$ , so its volume is $2\pi r^3$ , and $\tfrac{4}{3}\pi r^3 \div 2\pi r^3 = \tfrac{2}{3}$ .

Worked example: A basic sphere

Find the volume of a sphere with radius 3 cm, exactly and to the nearest tenth.

V = \tfrac{4}{3}\pi(3)^3 = \tfrac{4}{3}(27)\pi = 36\pi \approx 113.1 \text{ cm}^3

Common mistake

The radius is cubed, not squared. Writing $\tfrac{4}{3}\pi(3)^2 = 12\pi$ is a very common slip that comes from mixing up the volume formula with area formulas. Volume is measured in cubic units, so the formula must multiply three lengths.

A hemisphere is half of a sphere. Its flat face is a circle of radius r.

Worked example: A hemisphere from its diameter

A salad bowl is a hemisphere 12 in across. How much can it hold?

The radius is $12 \div 2 = 6$ in.

V = \tfrac{2}{3}\pi(6)^3 = \tfrac{2}{3}(216)\pi = 144\pi \approx 452.4 \text{ in}^3

Working backward

To find a radius from a volume, solve for $r^3$ , then take the cube root.

Worked example: Finding the radius

A sphere has a volume of $2304\pi$ cubic meters. Find its radius.

\begin{aligned} \tfrac{4}{3}\pi r^3 &= 2304\pi \\ r^3 &= 2304 \cdot \tfrac{3}{4} = 1728 \\ r &= \sqrt[3]{1728} = 12 \text{ m} \end{aligned}

Tip

Learn the first few perfect cubes: $2^3 = 8$ , $3^3 = 27$ , $4^3 = 64$ , $5^3 = 125$ , $6^3 = 216$ , $9^3 = 729$ , $10^3 = 1000$ , $12^3 = 1728$ . They show up constantly in sphere problems.

How sphere volume grows

Because the radius is cubed, sphere volumes grow fast. A ball that is only twice as wide as another holds eight times as much air, and a ball three times as wide holds 27 times as much. In general, multiplying the radius by $k$ multiplies the volume by $k^3$ . Doubling the radius gives $2^3 = 8$ times the volume, and cutting it in half leaves only $\tfrac{1}{8}$ of the volume.

This works in reverse, too. If one sphere holds 27 times as much as another, its radius is only $\sqrt[3]{27} = 3$ times as large. When a problem compares two spheres, you can often skip computing either volume and just compare the cubes of their radii.

Composite solids with spheres

Hemispheres often cap cylinders and cones. As with composite areas, add or subtract the pieces.

A capsule: a cylinder with a hemisphere on each end.

Worked example: A capsule

A capsule is a cylinder 10 cm long with a hemisphere of radius 3 cm on each end. Find its volume in terms of $\pi$ .

The two hemispheres together make one sphere of radius 3.

\begin{aligned} V &= \pi(3)^2(10) + \tfrac{4}{3}\pi(3)^3 \\ &= 90\pi + 36\pi \\ &= 126\pi \text{ cm}^3 \end{aligned}