A cube is a three-dimensional figure that has equivalent width, height, and length. This figure has six square faces, and all of its sides are of equal length, forming right angles. Finding the volume of a cube is easy – usually, just multiply your length × width × height. Since the sides of a cube are the same length, another way to think about volume is s 3, where s is the length of one of its sides. See Step 1 below for a more detailed analysis of these processes.
Method 1 of 3: Raising one side of the cube to the third power
Step 1. Find the length of one side of the cube
Generally, in problems that ask for the value of the volume of a cube, the length of one side is given. If you have access to this information, it is possible to calculate the cube's volume. If you want to find the volume in real life rather than a math exercise, use a ruler or measuring tape to calculate this measurement.
To better understand the process of calculating the volume of a cube, let's use an example by following the steps in this section. Let's imagine that the side of a cube measures 2 cm. This information will be used to calculate your volume in the next Step
Step 2. Raise the length from side to hub
When you find the value on the side of a cube, raise it to the third power. In other words, multiply it twice by itself. If s equals the length of the side, multiply s × s × s (or, more simply, s 3). The result will be the volume of the cube.
- This process is basically the same as finding the base area and multiplying it by the height (or, in other words, length × width × height), since the base area is found by multiplying its base by its height. Since the length, width, and height of a cube are equivalent, it is possible to shorten this process by raising any of these measurements to the third power.
- Let's continue with the example. Since the length of the side of the cube measures 2 cm, we can multiply 2 x 2 x 2 (or 23) =
Step 3. Identify the answer in cubic units
Since volume is the measure of three-dimensional space, the answer must be in cubic units by definition. Often, forgetting to put the unit of measure in math exercises can cause you to lose points, so pay attention to this detail.
- In the example used, as the original measurement is in centimeters, the final answer will be identified with the unit "cubic centimeters" (or in 3). Therefore, the answer "8" will become represented by 8 in3.
- The final answer will always be indicated according to the measure used initially. For example, if the measurement of the side of the cube were 2 "meters" - instead of 2 cm -, the final answer would be in cubic meters (m3).
Method 2 of 3: Calculating Volume from Surface Area
Step 1. Calculate the surface area of the cube
Although the easiest way to calculate the volume of a cube is to raise the length of one of its sides to the third power, it is not the only way. The length of one side of the cube or the area of one of its faces can be calculated from several other properties of this figure, which means that, by knowing some of this information, it is possible to calculate the cube's volume indirectly. For example, if you know the surface area value of the cube, all you need to do to calculate the volume is divide the surface area by 6 and then calculate the square root of that value to find the length of one side of the cube. cube. Then, just raise the length of the lateral to the third power to calculate the volume. This section presents a step by step of this process.
- The surface area of a cube is obtained by the formula 6 s 2, where s equals the length of one side of the cube. This formula is much the same as calculating the two-dimensional area of the six faces of a cube and adding these values together. Let's use it to calculate the cube's volume from its surface area.
- As an example, imagine a cube whose surface we know measures 50 cm2, but we do not know the value of the length of its side. In the next steps, we will use this information to calculate your volume.
Step 2. Divide the surface area of the cube by 6
Since the cube has 6 faces with an equal area, dividing its area by 6 results in the area of one of its faces. This area is equal to the lengths of its two sides multiplied (l × w, w × h or h × l).
- In our example, divide 50/6 = 8, 33 cm2. Don't forget that a two-dimensional answer has square units (cm2m2, and so on).
Step 3. Take the square root of this value
Since the area of one of the cube faces is equal to s 2 (s × s), taking the square root of this value gives the length of one side of the cube. After taking this measurement, you will have enough information to calculate the volume value as you normally would.
- In the example used, √8, 33 = 2.89 cm.
Step 4. Raise this value to the third power to find the cube's volume
Now that we know the value of the cube's side length, we simply raise it to the third power (multiply it twice by itself) to find the cube's volume as described in the section above. Congratulations – you have calculated the volume of a cube from its surface area.
- In the example used, 2, 89 × 2, 89 × 2, 89 = 24, 14 cm3. Don't forget to use the unit of measure to identify the answer.
Method 3 of 3: Calculating the volume from the diagonals
Step 1. Divide the diagonal of one of the cube faces by √2 to calculate the side length
By definition, the diagonal of a perfect square equals √2 × the length of one of its sides. So, if you only know the diagonal value of one of the cube's faces, you can calculate the value of its side by dividing the diagonal by √2. Then, the process for calculating the volume is relatively simple, as described in the Steps above.
- For example, let's say that one of the cube faces has a diagonal of 7 meters of lenght. To calculate the value of the side of the cube, divide 7/√2 = 4.96 meters. It is now possible to calculate the volume by multiplying 4, 963 = 122, 36 meters3.
- Note that, in general terms, d 2 = 2 s 2 where d is the diagonal length of one of the cube's faces, and s is the length of one of the sides. This is because, according to the Pythagoras Theorem, the square of the hypotenuse of a right triangle equals the sum of the squares of the other two sides. Thus, as the diagonal of one of the cube's faces and two sides of that face form a right triangle, d 2 = s 2 + s 2 = 2 s 2.
Step 2. Square the two opposite corners of the cube diagonally, then divide by 3 and take the square root to calculate the length of the side
If the only information you have about a cube is the length of a three-dimensional line segment that runs diagonally from one corner of the cube to the opposite corner, you can still calculate volume. Since d forms one side of a right triangle that has the diagonal between the two opposite corners of the cube as the hypotenuse, we can say that D 2 = 3 s 2, where D = is the three-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 D 2 = d 2 + s 2. As we found out earlier that d 2 = 2 s 2, we can say that D 2 = 2 s 2 + s 2 = 3 s 2.
- As an example, let's say we know that the diagonal from one corner of the cube's base to the opposite corner at the top of the cube is 10 m. If you want to calculate the volume, just use 10 in place of D in the equation above, as follows.
- D 2 = 3 s 2.
- 102 = 3 s 2.
- 100 = 3 s 2
- 33, 33 = s 2
- 5, 77 m = n. Then, just raise the length of the lateral to the third power to calculate the cube's volume.
- 5, 773 = 192, 45 m3