Surface Area of Cubes and Rectangular Prisms | Grade 7 Math | LevelKite

Think of a closed cardboard box. Suppose you want to cover its whole outside with paper: the top, the bottom and all four sides. The amount of paper you need is the box's surface area.

A box is a solid, which is a three-dimensional shape. Each flat outside surface of the box is called a face. The surface area of the box is the total area of all its faces, including the faces you cannot see when the box rests on a table.

In the last lesson, you found surface area by adding the areas of the faces in a net. The idea is the same here: find the area of every face, then add. Now you will work straight from the measurements.

When every length is a positive number in the same unit, such as centimeters, each face area is in the matching square unit, square centimeters ( $\text{cm}^2$ ). The total is in that square unit too.

Start with the simplest kind of box. A cube is a solid with six square faces that are all congruent, which means they have exactly the same shape and size. An edge is a line segment where two faces meet, and every edge of a cube has the same length.

Let $s$ stand for the edge length of a cube. It is a positive number in a unit of length. Each face is a square that measures $s$ by $s$ , so the area of one face is $s \times s = s^2$ , in the matching square unit.

Let $S$ stand for the surface area of the cube. All six faces have the same area, including the face the cube rests on and the faces turned away from you. So $S$ is six of those equal areas added together, which is $6$ times the area of one face:

S = 6 \times s^2

Worked example

Surface area of a cube

Every edge of a cube is $4$ cm long. What is the surface area of the cube?

Name the measure and the faces it describes. The edge length is 4 cm, a positive length. Each of the six faces is a square that is 4 cm by 4 cm.
$s = 4 \text{ cm}$
Find the area of one face. The area of a square is its side length times itself. Centimeters times centimeters gives square centimeters.
$4 \times 4 = 16 \text{ cm}^2$
Add the areas of all six faces. The faces are congruent, so adding six equal areas is the same as multiplying one area by 6. This counts the bottom face and the faces turned away from you.
$6 \times 16 = 96 \text{ cm}^2$

Answer

The surface area of the cube is $96$ square centimeters, or $96 \text{ cm}^2$ .

Many boxes are not cubes. A rectangular prism is a box-shaped solid with six faces that are all rectangles. Call the face it rests on the base. Three measurements, called its dimensions, describe the prism.

The length $l$ and the width $w$ are the two side lengths of the base.
The height $h$ is the distance from the base to the top, measured perpendicular to the base. Perpendicular means at a right angle, so the height goes straight up and never along a slant.

All three dimensions are positive numbers in the same unit of length.

Now match each face to its dimensions. The top is a copy of the bottom, so both measure $l$ by $w$ . The front and the back each run along the length and up the height, so both measure $l$ by $h$ . The left side and the right side each run along the width and up the height, so both measure $w$ by $h$ .

So the six faces come in three matching pairs. The two faces in a pair are congruent, so they have equal areas. You already know that the area of a rectangle is the product of its two side lengths. The next table uses that fact to give the area of each pair.

The six faces of a rectangular prism, in three matching pairs

Pair of faces	Each face measures	Area of one face	Area of the pair
Top and bottom	$l$ by $w$	$l \times w$	$2 \times l \times w$
Front and back	$l$ by $h$	$l \times h$	$2 \times l \times h$
Left and right	$w$ by $h$	$w \times h$	$2 \times w \times h$

Key concept

Add the areas of all six faces

The surface area $S$ of a rectangular prism is the sum of the areas of its six faces. The faces come in three matching pairs, so find three areas, double each one, and add.

This works when the length $l$ , width $w$ and height $h$ are positive lengths in one unit, with $h$ measured perpendicular to the base and never along a slant. Then $S$ is in the matching square unit. Square units measure area. Cubic units measure volume, the space inside a solid, which is a different quantity.

In the formula, $lw$ means $l \times w$ . A cube is the special case $l = w = h = s$ , so each pair has area $2s^2$ and $S = 2s^2 + 2s^2 + 2s^2 = 6s^2$ .

S = 2lw + 2lh + 2wh

Worked example

Surface area of a rectangular prism

A rectangular prism has a base that is $5$ cm long and $3$ cm wide. Its height, measured perpendicular to the base, is $2$ cm. What is the surface area of the prism?

Name each dimension and the faces it describes. The length is 5 cm and the width is 3 cm, so the top and bottom are each 5 cm by 3 cm. The height is 2 cm, so the front and back are each 5 cm by 2 cm, and the left and right sides are each 3 cm by 2 cm. All three dimensions are positive lengths in centimeters, so every area is in square centimeters.
$l = 5 \text{ cm}, \quad w = 3 \text{ cm}, \quad h = 2 \text{ cm}$
Find the area of the top and bottom together. One of these faces has an area of length times width. The bottom is congruent to the top, so double that area.
$2 \times 5 \times 3 = 2 \times 15 = 30 \text{ cm}^2$
Find the area of the front and back together. One of these faces has an area of length times height. The back is congruent to the front, so double that area.
$2 \times 5 \times 2 = 2 \times 10 = 20 \text{ cm}^2$
Find the area of the left and right sides together. One of these faces has an area of width times height. The two sides are congruent, so double that area.
$2 \times 3 \times 2 = 2 \times 6 = 12 \text{ cm}^2$
Add the three pairs. The six faces do not overlap, and together they cover the whole outside of the prism, so their areas add up to the surface area.
$30 + 20 + 12 = 62 \text{ cm}^2$

Answer

The surface area of the prism is $62$ square centimeters, or $62 \text{ cm}^2$ .

That fix gives you a second way to organize the same work. Add the areas of one face from each pair first, then double the sum: $S = 2(lw + lh + wh)$ .

Doubling a sum doubles every part of it. This is the distributive property, so $2(lw + lh + wh) = 2lw + 2lh + 2wh$ . Both forms count the same six faces. Both hold for any rectangular prism whose dimensions $l$ , $w$ and $h$ are positive lengths in one unit, with $h$ perpendicular to the base. Use the form that helps you keep track of every face.

Worked example

Covering a closed box

A closed box is a rectangular prism. Its base is $30$ cm long and $12.5$ cm wide. Its height, measured perpendicular to the base, is $8$ cm. You want to cover the whole outside with paper, including the bottom, with no overlap. How much paper do you need?

Match each dimension to its faces. The length is 30 cm and the width is 12.5 cm, so the top and bottom are each 30 cm by 12.5 cm. The height is 8 cm, so the front and back are each 30 cm by 8 cm, and the left and right sides are each 12.5 cm by 8 cm. All three dimensions are positive lengths in centimeters.
$l = 30 \text{ cm}, \quad w = 12.5 \text{ cm}, \quad h = 8 \text{ cm}$
Find the area of the top. The area of a rectangle is the product of its two side lengths. Splitting 12.5 into 12 plus 0.5 does not change the product and makes it easier to find.
$30 \times 12.5 = 30 \times 12 + 30 \times 0.5 = 360 + 15 = 375 \text{ cm}^2$
Find the area of the front and the area of one side in the same way. For the side, 12 times 8 is 96 and 0.5 times 8 is 4.
$30 \times 8 = 240 \text{ cm}^2, \quad 12.5 \times 8 = 96 + 4 = 100 \text{ cm}^2$
Add the three areas. This sum covers one face from each pair, so it is half of the surface area.
$375 + 240 + 100 = 715 \text{ cm}^2$
Double the sum. Each face you counted has a congruent partner on the opposite side of the box, so doubling counts all six faces, including the bottom.
$2 \times 715 = 1430 \text{ cm}^2$

Answer

You need $1430$ square centimeters of paper, or $1430 \text{ cm}^2$ , to cover all six faces of the box.

The takeaway: the surface area of a cube or a rectangular prism is the total area of all six faces, including the faces that are hidden.

A rectangular prism has six rectangular faces in three matching pairs: $l$ by $w$ , $l$ by $h$ and $w$ by $h$ . Its surface area is $2lw + 2lh + 2wh$ .
A cube has six congruent square faces that are each $s$ by $s$ . Its surface area is $6s^2$ .

Both rules need positive lengths in one unit, with the height perpendicular to the base. Both give an answer in square units.

Next, you will use the same face-by-face thinking to find the surface area of triangular prisms and pyramids.