Geometry of a Cube

A cube is a special type of rectangular prism where the length, width, and height are all the same. Or as lumen puts it: "A cube is a rectangular solid whose length, width, and height are equal." You can also think of a cube as a cardboard box made up of six equally sized squares.

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 much more easily calculate its geometry. The sections below demonstrate what formulas to use to find a cube's surface area and volume as well as geometric relationships within the shape.

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Surface Area of a Cube

Surface Area of A Cube
D. Russell

As noted, in a cube, all sides are the same. In the pictured example, you have a cube whose sides are represented as and h. You know that 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 (length) and (width). Since and are represented as 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 is 3 inches. You know that and 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.

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Volume of a Cube

Volume of a Cube
D. Russell

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
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Cube Relationships

Cube Relationships
D. Russell

Because you are working with a cube, you will have 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," notes 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.