The cube is inscribed in the ball (the vertices of the cube lie on the surface of the ball).

The cube is inscribed in the ball (the vertices of the cube lie on the surface of the ball). The surface of the cube is 18. Find the radius of the ball.

The diagonal of a cube inscribed in a ball is equal to the diameter of this ball.

All the faces of the cube are equal to each other, which means that the area of one face is 18/6 = 3.

The area of a cube face is equal to the square of the length of the cube’s edge:

S = a ^ 2;

a = √S = √3 – cube edge.

Knowing the length of the edge, by the Pythagorean theorem, we can find the diagonal of the face of the cube:

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

d = a√2 = √3 * √2 = √6 is the diagonal of the cube face.

We find the square diagonal of the cube as the sum of the squares of the edge and the diagonal of the face:

D ^ 2 = d ^ 2 + a ^ 2 = 6 + 3 = 9;

D = √9 = 3 is the diagonal of the cube.

The radius of a ball described about a cube is half the diagonal of the cube:

R = D / 2 = 3/2 = 1.5.