The area of the diagonal section of the cube is 4√2 cm ^ 2. Find the volume of a cube.

Let’s call the cube ABCDA1B1C1D1. Consider the section BB1DD1, but initially consider the right-angled triangle BAD (angle A = 90 degrees), denote each of the legs as x, then the hypotenuse will be equal to:
x ^ 2 + x ^ 2 = 2x ^ 2;
BD = √2x.
Let’s move on to the section, we denote BB1 as x, BD will be equal to √2x, then the cross-sectional area can be written as:
BB1 * BD = x * √2x = √2x ^ 2;
√2x ^ 2 = 4√2;
x ^ 2 = 4;
x = 2.
We got that the length of the edge of the cube is 2 cm, then the volume of the cube is:
V = BB1 ^ 3 = 2 ^ 3 = 8 cm3.
Answer: 8 cm3.