Find the volume of a cube if the area of its diagonal section is 2.

The base of the cube is a square. Its diagonal is the hypotenuse of a right-angled triangle, its two sides are legs. The square of the diagonal can be found as the sum of the squares of the two sides of the base:

d ^ 2 = a ^ 2 + a ^ 2 = 2 * a ^ 2;

d = a√2.

The area of the diagonal section is equal to the product of the diagonal of the base and the length of the edge of the cube:

Ssection = d * a = a√2 * a.

Knowing that the area of the diagonal section of the cube is 2, we can find the length of the edge of the cube:

a ^ 2 = Ssection / √2 = 2 / √2 = √2;

a = √ (√2) ≈ 1.189.

Cube volume:

V = a3 = (1.189) 3 ≈ 1.68.