Since we calculated above that d 2 = 2 s 2, we can say that D 2 = 2 s 2 + s 2 = 3 s 2. D, d, and s form a right triangle with D as the hypotenuse, so we can say that D 2 = d 2 + s 2. This is because of the Pythagorean Theorem.Because d forms one of the sides of a right triangle that has the diagonal between the two opposite corners of the cube as a hypotenuse, we can say that D 2 = 3 s 2, where D = the 3-dimensional diagonal between opposite corners of the cube. If the only piece of information you're given about a cube is the length of a 3-dimensional line segment stretching diagonally from one corner of the cube to the corner opposite it, it's still possible to find the volume of the cube. Square the diagonal of two opposite corners of the cube, then divide by 3 and take the square root to find the side length. Thus, because the diagonal of a cube's face and two of the sides on that face form a right triangle, d 2 = s 2 + s 2 = 2 s 2. 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. Note that, in general terms, d 2 = 2 s 2 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.Now that we know the side length, we can find the volume of the cube by multiplying 4.96 3 = 122.36 feet 3. We would find the side length of the cube by dividing 7/√2 = 4.96 feet. For instance, let's say that one of a cube's faces has a diagonal that is 7 feet long. From here, it's relatively simple to cube your answer and find the volume of the cube as described above. Thus, if the only information you're given about a cube is regarding the diagonal length of one of its faces, you can find the side length for the cube by dividing this value by √2. By definition, the diagonal of a perfect square is √2 × the length of one of its sides. In the next few steps, we'll use this information to find the cube's volume.ĭivide the diagonal across one of the cube's faces by √2 to find the cube's side length.
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