Math Core

How much cardboard does it take to make a box, and how much can the box hold? The first question is about surface area and the second is about volume. For any prism, both come from shapes you already know how to measure.

Prisms and their bases

A prism is a solid with two identical, parallel faces called bases, joined by rectangles. The prism is named for the shape of its bases: a rectangular prism has rectangle bases, and a triangular prism has triangle bases. The distance between the two bases is the prism's height (sometimes called its length when the prism lies on its side).

A rectangular prism 6 cm long, 4 cm deep and 3 cm tall.

Volume

Volume counts how many unit cubes fill a solid. Picture filling the box above with $1$ -centimeter cubes. The bottom layer is $6$ by $4$ , so it holds $24$ cubes, which is the area of the base. The box is $3$ layers tall, so it holds $24 \times 3 = 72$ cubes.

That idea works for every prism, whatever shape its base is.

Volume of a prism

V = Bh

$B$ is the area of the base and $h$ is the height of the prism (the distance between the bases). Volume is measured in cubic units, like $\text{cm}^3$ .

Surface area

Surface area is the total area of all the faces, like the amount of wrapping paper needed to cover the solid with no overlap. Unfold the prism into a flat net and add up the areas of the pieces.

A rectangular prism has three pairs of matching faces. For the box above:

top and bottom: $6 \times 4 = 24$ each
front and back: $6 \times 3 = 18$ each
left and right: $4 \times 3 = 12$ each

SA = 2(24) + 2(18) + 2(12) = 48 + 36 + 24 = 108 \text{ cm}^2

Worked example: A triangular prism

A tent-shaped prism has right-triangle bases with legs $3$ cm and $4$ cm and a longest side of $5$ cm. The prism is $10$ cm long. Find its volume and surface area.

A triangular prism. Each base is a right triangle with sides 3, 4 and 5 cm, and the prism is 10 cm long.

Volume. The base is a triangle with area $\dfrac{1}{2} \times 3 \times 4 = 6 \text{ cm}^2$ . So $V = Bh = 6 \times 10 = 60 \text{ cm}^3$ .

Surface area. There are two triangle bases and three rectangles, one for each side of the triangle:

\begin{aligned} \text{two triangles:}\ & 2 \times 6 = 12 \\ \text{three rectangles:}\ & 3 \times 10 + 4 \times 10 + 5 \times 10 = 120 \\ SA &= 12 + 120 = 132 \text{ cm}^2 \end{aligned}

Tip

The rectangles around a prism always have the prism's height as one side, and together their other sides go once around the base. So their total area is (perimeter of base) $\times$ height. In the example: $(3 + 4 + 5) \times 10 = 120$ .

Worked example: A missing dimension

A fish tank shaped like a rectangular prism holds $3{,}600$ cubic inches of water when full. Its base is $30$ inches by $12$ inches. How tall is it?

The base area is $30 \times 12 = 360$ square inches. Using $V = Bh$ :

\begin{aligned} 3600 &= 360h \\ h &= 10 \text{ inches} \end{aligned}

Common mistake

In $V = Bh$ , $B$ is an area, not a side length. For a triangular prism, find the triangle's area first ( $\dfrac{1}{2} \times \text{base} \times \text{height}$ ). Multiplying all three side lengths of the triangle by the prism's length gives a wrong answer.

Practice

Practice 1

A rectangular prism is $5$ cm long, $3$ cm wide and $2$ cm tall. What is its volume, in cubic centimeters?