The cube is a three-dimensional geometric solid, whose height, width and depth measurements are identical. A cube is made up of 6 square faces with all equal sides and right angles. Calculating the volume of a cube is very simple, since generally you need to do this simple multiplication: length × width × height. Since the sides of a cube are all the same, the formula for calculating its volume can be the following L 3, where l represents the measurement of a single side of the solid. Continue reading the article to find out how to calculate the volume of a cube in different ways.
Steps
Method 1 of 3: Knowing the Length of a Side
Step 1. Find the side length of the cube
Often math problems that require you to calculate the volume of a cube give the length of one side. If you have this information, you have everything you need to do the calculations. If you are not struggling with an abstract math or geometry problem, but are trying to calculate the volume of a real physical object, use a ruler or measuring tape to measure the length of one of the sides.
To better understand the process to follow to calculate the volume of a cube, in the steps of this section, we will tackle an example problem. Let's assume we are examining a cube whose side measures 5 cm. In the following steps we will use this data to calculate its volume.
Step 2. Cube the side length
Once we have identified how much one side of a cube measures, we raise this value to the cube. In other words, we multiply this number by itself three times. If l represents the length of the side of the cube under consideration, we will have to perform the following multiplication: l × l × l (i.e. l 3). In this way we will get the volume of the cube in question.
- The process is essentially identical to that of calculating the area of the base of the solid and then multiplying it by its height and, given that the area of the base is calculated by multiplying length and width, in other words we will use the formula: length × width × height. Knowing that length, width, and height are equal in a cube, we can simplify the calculations by simply cubeing one of these measurements.
- Let's proceed with our example. Since the length of one side of the cube is 5 cm, we can calculate its volume by performing this calculation: 5 x 5 x 5 (i.e. 53) = 125.
Step 3. Express the final result with a cubic unit of measurement
Since the volume of an object measures its three-dimensional space, the unit of measurement that expresses this size must be cubic. Often, not using the correct units of measurement during the math tests or checks that are faced in the school environment, you get lower scores or grades, so it is good to pay the utmost attention to this aspect.
- In our example, the initial measurement of the side of the cube is expressed in cm, so the final result we have obtained must be expressed in "cubic centimeters" (ie cm3). At this point, we can say that the volume of the studied cube is equal to 125 cm3.
- If we had used a different initial unit of measure, the final result would have changed. For example, if the cube had a side of 5 meters in length, instead of 5 centimeters, we would have obtained a final result expressed in cubic meters (i.e. m3).
Method 2 of 3: Knowing the Surface Area
Step 1. Find the surface area of the cube
While the simplest way to calculate the volume of a cube is to know the length of one of its sides, there are other ways to do the same. The length of one side of the cube or the area of one of its faces can be calculated starting from other quantities of this solid. This means that, knowing one of these two data, it is possible to calculate its volume using inverse formulas. For example, let's assume we know the surface area of a cube; starting from this data all we have to do to go back to its volume is to divide it by 6 and calculate the square root of the result, thus obtaining the length of a single side. At this point, we have everything we need to calculate the volume of a cube in the traditional way. In this section of the article we will go through the described process step by step.
- The surface area of a cube is calculated using the formula 6 l 2, where l represents the length of one of the sides of the cube. This formula is equivalent to calculating the surface area of each of the 6 faces of the cube and adding together the results obtained. Now we can use this formula, or rather the various inverse formulas, to calculate the volume of a cube starting from its surface area.
- For example, let's assume we have a cube whose total surface area is equal to 50 cm2, but of which we do not know the length of the sides. In the next steps of this section we will illustrate how to use this information to derive the volume of the cube under consideration.
Step 2. Let's start by dividing the surface area by 6
Since a cube is composed of 6 identical faces, to obtain the area of one of them, simply divide the total surface area by 6. The area of a face of a cube is obtained by multiplying the lengths of two of the sides that compose it (length × width, width × height or height × length).
In our example we will divide the total area by the number of faces to get 50/6 = 8.33 cm2. Remember that square units are always used to express a two-dimensional area (cm2, m2 and so on).
Step 3. We calculate the square root of the result obtained
Knowing that the area of one of the faces of the cube is equal to l 2 (i.e. l × l), calculating the square root of this value gives the length of a single side. Once this value has been obtained, we have all the information necessary to solve our problem in the classic way.
In our example we will get √8, 33 = 2, 89 cm.
Step 4. Cube the result
Now that we know how much a single side of our cube measures, to calculate its volume we will simply have to cube that measure (i.e. multiply it by itself three times), as shown in detail in the first section of the article. Congratulations, you are now able to calculate the volume of a cube from its total surface area!
In our example we will get 2, 89 × 2, 89 × 2, 89 = 24, 14 cm3. Do not forget that volumes are three-dimensional quantities, which must therefore be expressed with cubic units of measurement.
Method 3 of 3: Knowing the Diagonals
Step 1. Divide the length of one of the diagonals of the cube faces by √2, thus obtaining the measurement of a single side
By definition, the diagonal of a square is calculated as √2 × l, where l represents the length of one side. From here we can deduce that if the only information you have available is the length of a diagonal of a face of the cube, it is possible to find the length of a single side by dividing this value by √2. Once the measurement of one side of our solid has been obtained, it is very simple to calculate its volume as described in the first section of the article.
- For example, assume we have a cube whose diagonal of one face measures 7 meters. We can calculate the length of a single side by dividing the diagonal by √2 to get 7 / √2 = 4, 96 meters. Now that we know the size of one side of our cube, we can easily calculate its volume as follows 4, 963 = 122, 36 meters3.
- Note: In general terms, the following equation d holds 2 = 2 l 2, where d is the length of the diagonal of one of the faces of the cube and l is the measure of one of the sides. This formula is valid thanks to the Pythagorean theorem, which states that the hypotenuse of a right triangle is equal to the sum of the squares constructed on the two sides. Since the diagonal is nothing other than the hypotenuse of the triangle formed by the two sides of a face of the cube and by the diagonal itself, we can say that d 2 = l 2 + l 2 = 2 l 2.
Step 2. Even knowing the internal diagonal of a cube it is possible to calculate its volume
If the only data available to you is the length of the internal diagonal of a cube, that is the segment that connects two opposite corners of the solid, it is still possible to find its volume. In this case, it is necessary to calculate the square root of the internal diagonal and divide the result obtained by 3. Since the diagonal of one of the faces, d, is one of the legs of the right triangle which has the internal diagonal of the cube as its hypotenuse, we can say that D 2 = 3 l 2, where D is the internal diagonal joining two opposite corners of the solid and l is the side.
- This is always true thanks to the Pythagorean theorem. Segments D, d and l form a right triangle, where D is the hypotenuse; therefore, based on the Pythagorean theorem, we can say that D 2 = d 2 + l 2. Since in the previous step we stated that d 2 = 2 s 2, we can simplify the starting formula in D 2 = 2 l 2 + l 2 = 3 l 2.
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For example, let's assume that the internal diagonal of a cube connecting one of the corners of the base with the respective opposite corner of the top face measures 10 m. If we need to calculate its volume, we must substitute the value 10 for the variable "D" of the equation described above, obtaining:
- D. 2 = 3 l 2.
- 102 = 3 l 2.
- 100 = 3 l 2
- 33, 33 = l 2
- 5, 77 m = l. Once we have the length of a single side of the cube in question, we can use it to go back to the volume by raising it to the cube.
- 5, 773 = 192, 45 m3
