Math Core

Soup cans, water tanks, candles and drinking glasses are all cylinders. To know how much one holds, you need its volume. Good news: you already know the idea behind it from rectangular prisms. You just need a new base.

From prisms to cylinders

For any prism, the volume is the area of the base times the height:

V = Bh

Think of it as stacking layers. The base is one thin layer, and you stack $h$ units of those layers on top of each other.

A cylinder works the same way. It has two parallel, identical circles for bases, joined by a curved side. Every slice parallel to the base is the same circle, so the volume is still "base times height." The only change is that the base is a circle, and the area of a circle is $\pi r^2$ .

A cylinder with radius r and height h. The height is the distance between the two circular bases.

Volume of a cylinder

V = \pi r^2 h

where $r$ is the radius of the base and $h$ is the height (the distance between the bases). The volume is measured in cubic units.

Exact answers and approximate answers

Because $\pi$ is irrational, you can report a volume in two ways.

In terms of $\pi$ (exact): leave $\pi$ as a symbol, like $45\pi$ cubic inches.
As a decimal (approximate): multiply by $3.14$ or use the $\pi$ key on a calculator, then round, like $141.4$ cubic inches.

Always read the question to see which one it wants. An answer in terms of $\pi$ is exact, so it is often the cleaner choice.

Worked example: Using the radius

A cylinder has a radius of 3 inches and a height of 5 inches. Find its volume in terms of $\pi$ , then approximate it to the nearest tenth.

\begin{aligned} V &= \pi r^2 h \\ &= \pi \cdot 3^2 \cdot 5 \\ &= \pi \cdot 9 \cdot 5 \\ &= 45\pi \end{aligned}

The exact volume is $45\pi$ cubic inches. With a calculator, $45\pi \approx 141.4$ cubic inches.

Worked example: Starting from the diameter

A can is 10 cm across (its diameter) and 4 cm tall. Find its volume. Use $3.14$ for $\pi$ .

The formula needs the radius, which is half the diameter: $r = 10 \div 2 = 5$ cm.

\begin{aligned} V &= \pi \cdot 5^2 \cdot 4 \\ &= 100\pi \\ &\approx 100 \times 3.14 = 314 \end{aligned}

The can holds about $314$ cubic centimeters.

Working backward

Sometimes you know the volume and need a missing measurement. Put what you know into $V = \pi r^2 h$ and solve the equation.

Worked example: Finding a missing height

A cylinder has a volume of $72\pi$ cubic feet and a radius of 3 feet. How tall is it?

\begin{aligned} \pi r^2 h &= V \\ \pi \cdot 3^2 \cdot h &= 72\pi \\ 9\pi h &= 72\pi \\ h &= \frac{72\pi}{9\pi} = 8 \end{aligned}

The cylinder is 8 feet tall. Check: $\pi \cdot 9 \cdot 8 = 72\pi$ . ✓

To find a missing radius, solve for $r^2$ first, then take the square root. For example, if $\pi r^2 \cdot 4 = 100\pi$ , then $r^2 = 25$ , so $r = 5$ (a length can't be negative).

Common mistake

Two mistakes cause most wrong answers:

Using the diameter as the radius. If a problem gives the distance across the circle, divide by 2 first.
Doubling instead of squaring. $r^2$ means $r \cdot r$ , not $2r$ . For $r = 3$ , $r^2 = 9$ , not 6.

Tip

Check your units. Radius and height are lengths (like cm), and multiplying three lengths gives cubic units (cm³). If your answer is in square units, you left something out.

Practice

Practice 1

A cylinder has a radius of 2 m and a height of 7 m. Find its volume in cubic meters. Give your answer in terms of $\pi$ (type pi for $\pi$ ).