To better understand the process of finding the volume of a cube, let’s follow along with an example problem as we go through the steps in this section. Let’s say the side of the cube is 2 inches (5. 08 cm) long. We’ll use this information to find the volume of the cube in the next step.
This 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 area of the base is found by multiplying its length and its width. Since the length, width, and height of a cube are equal, we can shorten this process by simply cubing any of these measurements. Let’s proceed with our example. Since the length of the side of our cube is 2 inches, we can find the volume by multiplying 2 x 2 x 2 (or 23) = 8.
In our example, since our original measurement was in inches, our final answer will be labelled with the units “cubic inches” (or in3). So, our answer of 8 becomes 8 in3. If we had used a different initial unit of measurement, our final cubic units would differ. For instance, if our cube had sides with lengths of 2 meters, rather than 2 inches, we would label it with cubic meters (m3).
The surface area of a cube is given via the formula 6s2, where s is the length of one of the cube’s sides. This formula is essentially the same as finding the 2-dimensional area of the cube’s six faces and adding these values together. We’ll use this formula to find the volume of the cube from its surface area. [7] X Research source As a running example, let’s say that we have a cube whose surface we know to be 50 cm2, but we don’t know its side lengths. In the next few steps, we’ll use this information to find the cube’s volume.
In our example, dividing 50/6 = 8. 33 cm2. Don’t forget that two-dimensional answers have square units (cm2, in2, and so on).
In our example, √8. 33 is roughly 2. 89 cm.
In our example, 2. 89 × 2. 89 × 2. 89 = 24. 14 cm3. Don’t forget to label your answer with cubic units.
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. Note that, in general terms, d2 = 2s2 where d is the length of the diagonal of one of the cube’s faces and s is the length of one of the sides of the cube. This is because, according to the Pythagorean theorem, the square of the hypotenuse of a right triangle is equal to the sums 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.
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. As an example, let’s say that we know that the diagonal from one of the corners in the base of the cube to the opposite corner in the “top” of the cube is 10 m. 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 = s2 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
