A cube is a special type of rectangular prism where the length, width, and height are all the same. You can also think of a cube as a cardboard box made up of six equally sized squares. Finding the area of a cube, then, is quite simple if you know the correct formulas.

Normally, to find the surface area or volume of a rectangular prism, you need to work with a length, width, and height that are all different. But with a cube, you can take advantage of the fact that all sides are equal to easily calculate its geometry and find the area.

### Key Takeaways: Key Terms

**Cube**: A rectangular solid on which the length, width, and height are equal. You need to know the length, height, and width to find the surface area of a cube.**Surface area:**The total area of the surface of a three-dimensional object**Volume:**The amount of space occupied by a three-dimensional object. It is measured in cubic units.

## Finding the Surface Area of a Rectangular Prism

Before working to find the area of a cube, it's helpful to review how to find the surface area of a rectangular prism because a cube is a special type of rectangular prism.

A rectangle in three dimensions becomes a rectangular prism. When all sides are of equal dimensions, it becomes a cube. Either way, finding the surface area and the volume require the same formulas.

Surface Area = 2(lh) + 2(lw) + 2(wh)

Volume = lhw

These formulas will allow you to find the surface area of a cube, as well as its volume and geometric relationships within the shape.

## Surface Area of a Cube

In the pictured example, the sides of the cube are represented as *L *and *h*. A cube has six sides and the surface area is the sum of the area of all of the sides. You also know that because the figure is a cube, the area of each of the six sides will be the same.

If you use the traditional equation for a rectangular prism, where *SA *stands for surface area, you would have:

SA=6(lw)

This means that the surface area is six (the number of sides of the cube) times the product of *l *(length) and *w *(width). Since *l *and *w *are represented as *L *and * h*, you would have:

SA= 6(Lh)

To see how this would work out with a number, suppose that *L* is 3 inches and *h *is 3 inches. You know that *L *and *h *have to be the same because, by definition, in a cube, all sides are the same. The formula would be:

*SA = 6(Lh)**SA = 6(3 x 3)**SA = 6(9)**SA = 54*

So the surface area would be 54 square inches.

## Volume of a Cube

This figure actually gives you the formula for the volume of a rectangular prism:

V = L x W x h

If you were to assign each of the variables with a number, you might have:

*L *= 3 inches

*W* = 3 inches

*h* = 3 inches

Recall that this is because all of the sides of a cube have the same measurement. Using the formula to determine the volume, you would have:

*V = L x W x h**V = 3 x 3 x 3**V = 27*

So the volume of the cube would be 27 cubic inches. Note also that since the sides of the cube are all 3 inches, you could also use the more traditional formula for finding the volume of a cube, where the "^" symbol means you are raising the number to an exponent, in this case, the number 3.

*V = s ^*3*V =*3 ^ 3 (which means*V = 3 x 3 x 3*)*V = 27*

## Cube Relationships

Because you are working with a cube, there are certain specific geometric relationships. For example, line segment *AB* is perpendicular to segment *BF*. (A line segment is the distance between two points on a line.) You also know that line segment *AB* is parallel to segment *EF*, something you can clearly see by examining the figure.

Also, segment *AE *and *BC* are skewed. Skew lines are lines that are in different planes, are not parallel, and do not intersect. Because a cube is a three-dimensional shape, line segments *AE *and *BC* are indeed not parallel and they do not intersect, as the image demonstrates.