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A cube is a three-dimensional cube of equal width, height, and length. A cube has six square faces, all of which have equal and perpendicular sides. Calculating the volume of a cube is simple – usually, you just need to calculate the length × width × height of the cube. Since the sides of a cube are all the same length, another way of the volume formula is s 3 , where s is the side length of the cube. See the detailed explanation of this calculation in Step 1 below.
Steps
Find the third power of one side of a cube
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Find the length of one side of the cube. Usually, when a problem asks you to find the volume of a cube, you will find the length of one side of the cube. Once you have this data, you are ready to find the volume of the cube. If you’re not solving a theoretical problem but are trying to find the volume of a real object with the shape of a cube, use a ruler or tape measure to measure the side of the cube.
- To better understand the process of calculating the volume of a cube, follow the process step by step through the following example. Assume the side of the cube is 2 cm . We will use this data to find the volume of the cube in the next step.
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The third power of side lengths. Once you’ve figured out the side lengths of the cube, triple this number up. In other words, multiply this number by itself twice. If s is the side length, you would calculate s × s × s (or, more simply, s 3 ). This formula will give the volume value of the cube!
- The process is essentially the same as finding the area of the base, and then multiplying it by the cube’s height (or, in other words, length × width × height), since the base area is found by multiplying length with bottom width. Since the length, width, and height of the cube are of equal length, we can shorten this process by the third power of the lengths of any of these sides.
- Let’s continue with the example above. Since the side length of the cube is 2 cm, we can find the volume by multiplying 2 x 2 x 2 (or 2 3 ) = 8 .
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Mark your answer with the triple power symbol. Since volume is a unit of measurement of three-dimensional space, the rule of thumb is that your answer should be in the form of the third power. Usually, in school math homework, if you don’t pay attention to write your answer in the correct unit, you will lose points, so don’t forget to use the correct units!
- In our example, since the initial unit of measure was cm, the final answer would have to be in “cubic centimeters” (or cm 3 ). So our answer 8 will become 8 cm 3 .
- If we use a different unit of measure at first, the final volume unit will also be different. For example, if our cube has a side of 2 meters , instead of 2 cm, we will write the unit as cubic meters (m 3 ).
Find the volume from the total area
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Find the total area of the cube. The easiest way to find the volume of a cube is to the cubic power of one of its sides, but that’s not the only way. The length of a side of a cube or the area of a face of a cube can be inferred from other properties of the cube, that is, if you start with one of these data, you can find the volume of the cube by the slightly longer path. For example, if you know the total area of a cube, all you need to do is divide the total area of the cube by 6, then take the square root of this value to find the length of the side. cube shape . From there, you just need to triple the side lengths to find the volume as usual. In this section, we will perform the calculation step by step.
- The total area of the cube is calculated by the formula 6 s 2 , where s is the length of the side of the cube. This formula is essentially similar to the formula for calculating the two-dimensional area of each face of a hexagon and adding these values together. We will use this formula to calculate the volume of a cube from its total area.
- For example, let’s say we have a cube with a total area of 50 cm 2 , but we don’t know the side lengths of the cube. In the next steps, we will use this data to find the volume of the cube.
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Divide the cube’s total area by 6. Since the cube has 6 faces with equal areas, dividing the cube’s total area by 6 gives you the area value of one face. This area is equal to the product of the two sides of a cube face (length × width, width × height, or height × length).
- In our example, we have the division 50/6 = 8.33 cm 2 . Don’t forget the answer to the area of a two-dimensional figure in square units (cm 2 , in 2 , and the like).
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Calculate the square root of this value. Since the area of one face of the cube is equal to s 2 ( s × s ), the square root of this will give you the length of the side of the cube. Once you have the side lengths of the cube, you have enough data to calculate the volume of the cube as usual.
- In our example, √8.33 = 2.89 cm .
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Triple this value to find the volume of the cube. Now that you have the value of the side length of the cube, triple it (multiply this value by itself) to find the volume of the cube as explained in detail above. . Congratulations! You have found the volume of a cube based on its total area.
- In our example, 2.89 × 2.89 × 2.89 = 24.14 cm 3 . Don’t forget to write your answer in blocks.
Find the volume from the diagonal
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Divide the diagonal of a cube face by √2 to find the length of the side of the cube. In principle, the diagonal of a square is equal to √2 × the length of one side of the square. Therefore, if the only information you have is about the diagonal of a cube face, you can find the length of the side of the cube by dividing the obtained value by √2. From then on, it was relatively simple to compute the third power of the side lengths and find the volume of the cube as described above.
- For example, suppose one face of a cube has a diagonal length of 2.13 meters . We will find the side length of the cube by dividing 2.13/√2 = 1.51 meters. Now that we know the side lengths, we can find the volume of the cube by multiplying 1.51 3 = 3.442951 m 3 .
- Note that, according to the general formula, d 2 = 2 s 2 where d is the length of the diagonal of a cube face and s is the length of the side of the cube. This is because, according to the Pythagorean theorem, the square of the hypotenuse of a right triangle is equal to the sum of the squares of the other two sides. Therefore, since the diagonal of a cube face and the two square sides of that face make a right triangle, d 2 = s 2 + s 2 = 2 s 2 .
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Square the diagonal drawn from two opposing points on the cube, then divide by 3 and calculate the square root of the value found to find the length of the side of the cube. If the only data you have about the cube is the diagonal in three-dimensional space drawn from this corner of the cube to the corner about it, you can still find the volume of the cube. Since d becomes a right-angled side of a right triangle with the hypotenuse being the diagonal between the two corners of the cube, we have D 2 = 3 s 2 , where D = the diagonal in three-dimensional space connecting the two opposite angles each other of the cube.
- This formula is derived from the Pythagorean Theorem. D , d , and s form a right triangle with D being the hypotenuse, so we have D 2 = d 2 + s 2 . As calculated above, d 2 = 2 s 2 , we have D 2 = 2 s 2 + s 2 = 3 s 2 .
- For example, suppose we know that the length of the diagonal from one corner on the base of the cube to its opposite corner on the “top” of the cube is 10 m. If we wanted to calculate the volume, we would substitute the value 10 for “D” in the above formula as follows:
- D 2 = 3 s 2 .
- 10 2 = 3 s 2 .
- 100 = 3 s 2
- 33.33 = s 2
- 5.77 m = s. From here, all we need to do to find the volume of the cube is the cubic power of the sides of the cube.
- 5.77 3 = 192.45 m 3
X
wikiHow is a “wiki” site, which means that many of the articles here are written by multiple authors. To create this article, 67 people, some of whom are anonymous, have edited and improved the article over time.
This article has been viewed 245,573 times.
A cube is a three-dimensional cube of equal width, height, and length. A cube has six square faces, all of which have equal and perpendicular sides. Calculating the volume of a cube is simple – usually, you just need to calculate the length × width × height of the cube. Since the sides of a cube are all the same length, another way of the volume formula is s 3 , where s is the side length of the cube. See the detailed explanation of this calculation in Step 1 below.
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