Math Core

Volume measures how much space a solid fills, in cubic units. Prisms and cylinders share one simple idea: they are the same shape all the way through. Once you see that, a single formula covers boxes, triangular tents, hexagonal pencils and soup cans.

Stacking layers

A prism has two congruent, parallel polygon faces called bases, joined by parallelogram faces. A cylinder is the same idea with circular bases. In a right prism or cylinder, the sides are perpendicular to the bases.

A rectangular prism. Hidden edges are dashed.

Think of a rectangular prism that is $l$ units long, $w$ units wide and $h$ units tall. The bottom layer holds $l \cdot w$ unit cubes, which is the area of the base, $B$ . There are $h$ layers, so the prism holds $Bh$ cubes.

Nothing in that argument depends on the base being a rectangle. Any prism or cylinder is a stack of thin layers, each a copy of the base. The volume is the area of one layer times the number of layers, or the base area times the height.

Volume of a prism or cylinder

V = Bh

where $B$ is the area of a base and $h$ is the height, the perpendicular distance between the bases. For a cylinder, $B = \pi r^2$ , so

V = \pi r^2 h

Common mistake

In $V = Bh$ , the letter $B$ is an area, not a length. For a triangular prism, $B$ is the area of the triangle, $\frac{1}{2}bh$ . A common error is to multiply the three side lengths you see. Find the area of the base first, then multiply by the height of the prism.

Triangular prisms

A triangular prism. Its bases are the two right triangles at the front and back.

A triangular prism is easy to misread because it often lies on one of its rectangular faces. The bases are the two triangles, wherever they happen to be. The "height" of the prism is the distance between the triangles, which here is its length.

Worked example: A triangular prism

A prism has right-triangle bases with legs of 5 in and 12 in. The prism is 9 in long. Find its volume.

The base is the right triangle, so its legs serve as base and height:

B = \tfrac{1}{2}(5)(12) = 30 \text{ in}^2

The distance between the triangles is 9 in:

V = Bh = 30 \cdot 9 = 270 \text{ in}^3

Cylinders

A cylinder with radius r and height h. The back half of the bottom circle is hidden, so it is dashed.

Worked example: A cylinder from its diameter

A can is 10 cm across and 12 cm tall. Find its volume in terms of $\pi$ and to the nearest cubic centimeter.

The radius is $10 \div 2 = 5$ cm.

\begin{aligned} V &= \pi r^2 h = \pi(5)^2(12) \\ &= 300\pi \\ &\approx 942 \text{ cm}^3 \end{aligned}

Cavalieri's principle

What about an oblique prism or cylinder, where the sides lean? Picture a neat stack of coins, then push the stack so it leans. Each coin is still the same coin, and there are still the same number of them, so the volume hasn't changed.

Seen from the side: a right prism and an oblique prism with the same base and the same height h.

This is the idea behind a principle named for the Italian mathematician Bonaventura Cavalieri.

Definition

Cavalieri's principle

If two solids have the same height, and every plane parallel to their bases cuts them in cross sections of equal area, then the two solids have the same volume.

An oblique prism and a right prism with the same base and the same height have matching cross sections at every level. So $V = Bh$ works for oblique prisms and cylinders too, as long as $h$ is the perpendicular height, not the length of the slanted side.

Worked example: An oblique cylinder

An oblique cylinder has a radius of 3 m. Its slanted side is 10 m long, and its perpendicular height is 8 m. Find its volume in terms of $\pi$ .

Use the perpendicular height, 8 m. The 10 m slant length does not matter.

V = \pi(3)^2(8) = 72\pi \text{ m}^3

Working backward

If you know a volume, you can solve $V = Bh$ for a missing dimension.

Worked example: How tall is the tank?

A cylindrical tank holds $567\pi$ cubic feet of water and has a radius of 9 ft. How tall is it?

\begin{aligned} \pi(9)^2 h &= 567\pi \\ 81h &= 567 \\ h &= 7 \text{ ft} \end{aligned}

Check: $81 \cdot 7 = 567$ . ✓

Tip

Watch how the dimensions scale. Doubling the height of a cylinder doubles its volume, but doubling the radius multiplies the volume by $2^2 = 4$ , because the radius is squared.