A cube is a three-dimensional figure with equal width, height, and length dimensions. The cube has six square faces, all of which have equal side lengths and all intersect at right angles. Finding the volume of a cube is not difficult at all-usually, all you need to do is multiply the length of the cube by its width and height. Since the lengths of all sides of the cube are equal, another way to represent the volume of the cube is s3, where s is the length of one side of the cube. For a detailed breakdown of these processes, see step 1 below.
- Cubing One of the Cube’s Sides
- Finding Volume from Surface Area
- Finding Volume from Diagonals
Method 1 of 3:
1. Cubing One of the Cube’s Sides
A. Determine the length of one side of the cube. Usually in tasks that ask you to find the volume of a cube, you will be given the length of one of the sides of the cube. If you have this information, you can resolve the volume of the cube. If you are not solving abstract mathematical problems, but trying to find the volume of a cube-shaped real object, use a ruler or tape measure to measure the sides of the cube.
- To better understand the cube volume determination process, let us follow the steps in this section to view an example problem. Assume that the sides of the cube are 2 inches (5.08 cm). We will use this information in the next step to find the cube volume.
B. Side length cube. When you find the length of one side of the cube, cube this number. In other words, double it yourself.If s is the length of the side, you would multiply s × s × s (or, in simplified form, s3).This will give you the volume of your cube!
- The process is essentially the same as finding the area of the bottom and then multiplying it by the height of the cube (or in other words, length × width × height), because the area of the bottom is determined by multiplying its length and width. Since the length, width and height of the cube are equal, we can shorten this process by simply adding the cube to any of these dimensions.
- Let’s continue our example. Since the side length of our cube is 2 inches, we can multiply it by 2 x 2 x 2 to find the volume(or 23) = 8.
C. Mark the answer in cubic units. Since volume is a measure of three-dimensional space, by definition, your answer should be in cubic units. Generally, in mathematics classes, ignoring the answer label and the correct unit will result in a decrease in the score of the question, so be sure to use the correct label!
- In our example, since our initial unit of measure is inches, the final answer will be marked in cubic inches (or inches3). So our answer is that 8 will become 8 inches3.
- If we use a different initial unit of measure, the final cubic unit will be different. For example, if the sides of our cube are 2 meters long instead of 2 inches long, then we will mark it as cubic meters (m3).
Method 2 of 3:
2. Finding Volume from Surface Area
A. Find the surface area of your cube. Although the simplest way to find the volume of a cube is to determine the length of the cube’s sides, this is not the only way. The length of the cube’s edges or the area of one of the faces can be obtained from some other properties of the cube, which means that if you start from one of these pieces of
information, you can find the volume of the cube in the solution.For instance, if you know a cube’s surface area, all you need to do to find its volume is to divide the surface area by 6, then take the square root of this value to find the length of the cube’s sides.
From here, you only need to specify the length of the side to find the volume as usual. In this section, we will gradually complete this process.
- The surface area of the cube is given by formula 6s2, where s is the length of one of the sides of the cube. In fact, this formula is the same as finding the two-dimensional area of the six faces of the cube and adding these values. We use this formula to find the volume of a cube by its surface area.
- For example, suppose we have a cube with a surface of 50 cm2, but we don’t know its side length. In the next few steps, we will use this information to find the volume of the cube.
B. Divide the surface area of the cube by 6. Since the cube has 6 equal faces, divide the surface area of the cube by 6 to get the area of one of the faces.This area is equal to the lengths of two of its sides multiplied (l × w, w × h, or h × l).
- In our example, dividing 50/6 = 8.33 cm2. Don’t forget that the unit of the two-dimensional answer is square (centimete2, inch2, etc.).
C. Take the square root of this value. Since the area of one face of the cube is s2(s × s), you will find the length of one face of the cube in units of the square root of this value, Now you have enough information to determine the volume of the cube as usual.
- In our example, √8.33 is roughly 2.89 cm.
D. The cube is used to find the value of the cube volume. Now that you have a value for the side length of the cube, just click this value (multiply yourself twice) to find the volume of the cube, as described in the previous section. Congratulations-you discovered the volume of the cube through the surface area of the cube.
- In our example, 2.89 × 2.89 × 2.89 = 24.14 cm3. Remember to mark your answers in cubic units.
Method 3 of 3:
3. Finding Volume from Diagonals
A. Divide the diagonal of any face of the diagonal by √2 to find the length of the opposite side. By definition, the diagonal of a perfect square is √2 × the length of one side. Therefore, if the only information you provide about the cube is about the diagonal length of one of its faces, you can find the side length of the cube by dividing this value by √2. From here, it is relatively easy to define a cube, answer and find the volume of the cube, as described above.
- For instance, let’s say that one of a cube’s faces has a diagonal that is 7 feet long. We would find the side length of the cube by dividing 7/√2 = 4.96 feet. Now that we know the side length, we can find the volume of the cube by multiplying 4.963 = 122.36 feet3 .
- Please note that in general d2 = 2s2 , where d is the diagonal length of one face of the cube and s is the length of one face of the cube. This is due to the fact that according to Pythagorean theorem, the square of the hypotenuse of a right triangle is the sum of the squares of the other two sides. Thus, because the diagonal of a cube’s face and two of the sides on that face form a right triangle, d2 = s2 + s2 = 2s2.
B. Diagonally align the two opposite corners of the cube, then divide by 3 and take the square root to find the length of the side. If the only information you provide about the cube is the length of a diagonal line from one corner of the cube to three diagonal segments, you can still find the volume of the cube. Because d forms one of the sides of a right triangle, the diagonal between the two opposite corners of the cube of this right triangle is like a hypotenuse, we can say that D2 = 3s2, where D = the 3-dimensional diagonal between opposite corners of the cube.
- This is because of the Pythagorean Theorem.D, d, and s form a right triangle with D as the hypotenuse, so we can say that D2 = d2 + s2. Since we calculated above that d2 = 2s2, we can say that D2 = 2s2 + s2 = 3s2.
- For example, suppose we know that the diagonal from an angle at the bottom of the cube to the opposite angle at the “top” of the cube is 10 meters. If we want to find the volume, we would insert 10 for each “D” in the equation above as follows:
- D2 = 3s2..
- 102 = 3s2.
- 100 = 3s2
- 33.33 = s
- 5.77 m = s. From here, all we need to do to find the volume of the cube is to cube the side length.
- 5.773 = 192.45 m3