How to Find the Volume of a Cube from Its Surface Area

Co-authored by David Jia

Last Updated: December 9, 2022

The volume of a three-dimensional shape is a measurement of the space inside the shape and is determined by multiplying the length, width, and height. A cube is a three-dimensional shape in which the length, width, and height are equal. Thus, it is easy to find the volume of a cube given the length of one edge. You can also find the volume using the surface area, from which you can derive the length of one edge.

Part 1

Part 1 of 2:

Finding the Length of One Edge

1
Set up the formula for the surface area of a cube. The formula is $surfacearea=6x^{2}$ $surfacearea=6x^{{2}}$ , where $x$ $x$ equals the length of one edge of the cube.^{[1]
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- To find the volume of a cube, you need to multiply its three dimensions (length, width, height) together.^{[2]
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  Research source} These dimensions correspond to the length of the edges of the cube. Since all the dimensions (edges) of a cube are the same, in order to find the volume of the cube, you first have to determine the length of one of its edges. Since finding the surface area of a cube also requires the length of one edge, if you know the surface area, you can work backwards to find the length of one edge, then use the edge length to work forward to find the volume.
2
Plug the surface area of the cube into the formula. This information should be given.
- If you do not know the surface area of the cube, this method will not work.
- If you already know the length of one edge of the cube, you can skip the following steps and plug in that value for $x$ into the volume of a cube formula: $volume=x^{3}$ .
- For example, if the surface area of your cube is 96 square centimeters, your formula will look like this:
  $96cm^{2}=6x^{2}$
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3
Divide the surface area by 6. This will give you the value of $x^{2}$ $x^{{2}}$ .
- For example, if the surface area of your cube is 96 square centimeters, you would divide 96 by 6:
  $96cm^{2}=6x^{2}$
  ${\frac {96}{6}}={\frac {6x^{2}}{6}}$
  $16=x^{2}$
4
Find the square root. This will give you the value of $x$ $x$ , or the length of one edge of the cube.
- You can find the square root using a calculator, or by hand. For complete instructions, read Calculate a Square Root by Hand.
- For example, if $16=x^{2}$ , then you need to find the square root of 16:
  $16=x^{2}$
  ${\sqrt {16}}={\sqrt {x^{2}}}$
  $4=x$
  So, the length of one edge for a cube with a surface area of $96cm^{2}$ is $4cm$ .

Part 2

Part 2 of 2:

Finding the Volume

1
Set up the formula for the volume of a cube. The formula is $v=x^{3}$ , where $v$ equals the volume of the cube, and $x$ equals the length of one edge.^{[3]
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2
Plug the length of one edge into the formula. You should have already calculated this from the given surface area.
- For example, if one edge of a cube is 4 centimeters, then your formula will look like this:
  $v=4^{3}$ .
3
Cube the length of one edge. To do this, you can use a calculator, or simply multiply x by itself three times. This will give you the volume of your cube, in cubic units.^{[4]
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- For example, if the length of one edge is 4 centimeters, you would calculate:
  $v=4^{3}$
  $v=4\times 4\times 4$
  $v=64$
  So, the volume of a cube with an edge length of 4 centimeters is $64cm^{3}$

Community Q&A

Question

What about a rectangular prism? Would it still be the same?

Community Answer

Not quite. The formula is slightly different. Assuming you know the length, width and height of the prism (l, w, and h), you would multiply the three values together to get the volume, or you would find twice the sum of (lw + lh + wh) to find the surface area. If you know only two of those values, but you also know either the volume or the surface area, you can work backwards to find the unknown value.
Question

The volume of a cube is 64 mm^3. How do I calculate the area of one of its faces?

Donagan

Top Answerer

The length of an edge is the cube root of 64, which is 4 mm. The face area is 4² = 16 sq mm.
Question

What is the volume of a cube with a surface area of 78 inches?

Community Answer

First you need to find the length of one edge using the formula for surface area: 78 = 6x^2 13 = x^2 3.61 inches = x Now that you know the length of one edge, you can plug this value into the formula for volume of a cube: v = 3.61^3 v = 3.61 x 3.61 x 3.61 v = 47.05 cubic inches