The diagonal of a cube is 5 cm. Find the area of one of its faces and the volume.

The diagonal of the cube is the hypotenuse in the right-angled triangle formed by this diagonal, the diagonal of one of the faces and the edge of the cube, so if a is the edge of the cube, b is the diagonal of one of the faces, then by the Pythagorean theorem:

5 ^ 2 = a ^ 2 + b ^ 2; (one)

At the same time, the diagonal of a cube face is a hypotenuse in a right-angled triangle formed by the diagonal of the face and two edges of the cube, thus, by the Pythagorean theorem:

b ^ 2 = a ^ 2 + a ^ 2; (2)

Substitute the expression in ^ 2 from (2) into (1):

25 = 3a ^ 2.

The area of ​​the edge of the cube is a ^ 2 = 25/3 = 8⅓ cm²; cube volume
a ^ 3 = (25/3) ^ (3/2) = 125 / (3√3) cm³.

Answer: the area of ​​the edge of the cube is 8⅓ cm²; cube volume 125 / (3√3) cm³.