How To Find The Surface Area Of A Rectangular Prism

Written by

Malcolm McKinsey

January 16, 2023

Fact-checked by

Paul Mazzola

What is a rectangular prism?

A rectangular prism is a six-faced, three-dimensional solid in which all the faces are rectangles. All six faces meet at right angles to one another. Opposite faces are congruent.

A special type of rectangular prism is a cube, in which all six faces are congruent.

What is the surface area of a rectangular prism?

The surface area of a rectangular prism is the total area of all six faces. When you have a cube, finding the area of one face allows you to find the total surface area of the solid very quickly, since it will be six times the area of one face.

Finding the surface area of all rectangular prisms allows you to also find the surface area of any cube, since a cube is a type of rectangular prism.

Surface area of a rectangular prism formula

Finding surface area for all rectangular prisms (including cubes) involves both addition and multiplication. You must know the width, length and height of the prism before you can apply this formula:

In this formula we have abbreviations for width (w), length (l) and height (h), and we can simplify that by factoring out the 2:

Since every face of a rectangular prism has a congruent, opposite face, you are tracking down all six faces in pairs. Using the formula helps prevent confusion or keeping track of which faces you have measured. You only need the three dimensions.

Surface area of a rectangular box formula

For a cube or rectangular box, the formula becomes even easier. Take the length of any edge, a:

This works because every dimension of a cube - width, height, and length - is the same. Any two measurements will give the area of one face, and the cube has six faces, so the area is $6{a}^{2}$.

If you have trouble remembering that special formula, you can always use the general one for rectangular prisms.

How to find the surface area of a rectangular prism

You have been asked to wrap a gift box your mathematics club will give to your math club adviser. The box contains $\sqrt{100}$ books of math jokes, so it is a good-sized rectangular prism. (Do you know how many books are in the box? 😊)

[insert cartoon of box with labeled dimensions as shown]

Its dimensions are:

• Width - 30 cm

• Length - 15 cm

• Height - 20 cm

Use the area formula to find out the minimum amount of gift wrap you will need. Work first; then peek.

Let's build our equation for surface area of a rectangular prism starting with our formula:

Though that sounds like a lot of gift wrap, it is only $0.27{m}^{2}$. You have a sheet of gift wrap that is 0.75m × 0.5m. Do you think you will have enough?

Sure, even if you leave a little extra for overlap, since you have $0.375{m}^{2}$ and only need $0.27{m}^{2}$!

Surface area of a rectangular box

Now lets look at how to find the surface area of a right rectangular prism, box, or cube.

You also have to wrap the math club's vaunted Cube Root Cube for summer storage. The cube has 12 congruent edges, each 45 cm45 cm. Work first; then peek.

[insert cartoon of fancy box labeled Cube Root Cube]

Let's build our equation for surface area of a rectangular box starting with our cube formula:

As is your club's tradition, you will use the Permanent Records of the Club Elders to wrap the prized Cube Root Cube. You have $2{m}^{2}$ of their records dating back to 1960. Though $\mathrm{12,150}c{m}^{2}$ sounds like a lot, it is only $1.215{m}^{2}$, so you have plenty of storage wrap to protect the Cube and its priceless contents of cube roots.

How to calculate surface area examples

Practice using these surface area word problems for rectangular prisms and cubes. Before looking at the answers, try your hand at both! See if you get the right answers.

You need to cover the travel cage for your pet boa constrictor so you can carry it on the school bus. The cage is 1.5 yards long, 0.25 yards wide and 0.25 yards high.

What is the surface area of the cage?

Let's try another word problem to find the surface area of a rectangular box. Your mathematics teacher is shipping a box full of imaginary numbers to a colleague across the country and has asked you to wrap it in brown kraft paper. The box is 12"x12"x12".

How much paper do you need?

You need $864i{n}^{2}$ of paper, which is only six square feet!