Math Core

How much cereal fits in a box, and how much cardboard does it take to make the box? The first question is about volume, the space inside a solid. The second is about surface area, the total area of all its outside faces. They measure different things, so they use different units.

Volume of a prism

A prism is a solid with two identical, parallel bases joined by rectangles. Think of stacking identical layers: the base is one layer, and the height tells you how many layers there are.

Volume of a prism or cylinder

V = Bh

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

For a rectangular prism, $B = \ell w$ , so $V = \ell w h$ .

A rectangular prism 5 in long, 2 in wide and 3 in tall. Dashed edges are hidden.

Worked example: A rectangular prism

Find the volume of the box above.

V = \ell w h = 5 \times 2 \times 3 = 30 \text{ in}^3

You could fit $30$ one-inch cubes inside: $10$ in the bottom layer, and $3$ layers.

Worked example: A triangular prism

A tent is a prism whose bases are triangles with base $6$ ft and height $4$ ft. The tent is $9$ ft long. Find its volume.

The base is the triangle: $B = \dfrac{1}{2}(6)(4) = 12 \text{ ft}^2$ . The "height" of the prism is the distance between the two triangles, $9$ ft.

V = Bh = 12 \times 9 = 108 \text{ ft}^3

Volume of a cylinder

A cylinder is like a prism with circles for bases. The same rule works: base area times height. The base is a circle, so $B = \pi r^2$ .

V = \pi r^2 h

A cylinder with radius 3 cm and height 10 cm.

Worked example: A cylinder

Find the volume of the cylinder above, in terms of $\pi$ and using $3.14$ .

V = \pi r^2 h = \pi(3^2)(10) = 90\pi \approx 282.6 \text{ cm}^3

Surface area

Imagine cutting a box along some edges and flattening it. The flat pattern is called a net. The surface area is the total area of the net.

A rectangular prism has $6$ faces in $3$ matching pairs: top and bottom, front and back, left and right.

SA = 2\ell w + 2\ell h + 2wh

A cylinder's net is two circles and one rectangle. The rectangle wraps around the side, so its length is the circumference $2\pi r$ and its width is the height $h$ .

SA = 2\pi r^2 + 2\pi r h

Worked example: Surface area of the box

Find the surface area of the $5$ in by $2$ in by $3$ in box.

Top and bottom: $2(5 \times 2) = 20$
Front and back: $2(5 \times 3) = 30$
Left and right: $2(2 \times 3) = 12$

$SA = 20 + 30 + 12 = 62 \text{ in}^2$ .

Common mistake

Don't mix up volume and surface area. Volume multiplies three lengths and gives cubic units. Surface area adds areas of faces and gives square units. The box above has volume $30 \text{ in}^3$ but surface area $62 \text{ in}^2$ .

Tip

When you know the volume and all but one dimension, write $V = Bh$ as an equation and solve for the missing length, just like any other equation.

Practice

Practice 1

A rectangular prism is $8$ cm long, $4$ cm wide and $3$ cm tall. What is its volume, in cubic centimeters?