You will need

- the length of a side of a cube, the radius of the inscribed and circumscribed ball

Instruction

1

The volume of a box is: V = abc, where a, b, c is its dimension. Therefore, the volume

**of a cube**is equal to V = a*a*a = a^3, where a is the side length**of the cube**.The surface area of**a cube**is equal to the sum of the areas of all its faces. Only**the cube**has six faces, so**the area of**its surface equal to S = 6*(a^2).2

Let the ball inscribed in the cube. Obviously, the diameter of the ball is equal to the side

**of the cube**. Substituting the length of the diameter in the expression for the volume instead of the length of the edges**of the cube**and using that the diameter is twice the radius, then get V = d*d*d = 2r*2r*2r = 8*(r^3) where d is the diameter of the inscribed circle, and r is the radius of the inscribed circle.The surface area of**the cube**will then be equal to S = 6*(d^2) = 24*(r^2).3

Let the ball circumscribed around

Consider first one of the faces

**the cube**. Then its diameter will coincide with the diagonal**of the cube**. The diagonal**of the cube**passes through the center**of the cube**and connects two opposite points.Consider first one of the faces

**of the cube**. The edges of this face are the legs of a right triangle in which the face diagonal d is the hypotenuse. Then by the Pythagorean theorem we get: d = sqrt((a^2)+(a^2)) = sqrt(2)*a.4

Then consider the triangle in which the hypotenuse is the diagonal

So, derived the formula of diagonal

**of a cube**and the diagonal edge of d and one edge**of cube**a to his legs. Similarly, by the Pythagorean theorem we get: D = sqrt((d^2)+(a^2)) = sqrt(2*(a^2)+(a^2)) = a*sqrt(3).So, derived the formula of diagonal

**of a cube**is equal to D = a*sqrt(3). Hence, a = D/sqrt(3) = 2R/sqrt(3). Therefore, V = 8*(R^3)/(3*sqrt(3)), where R is the radius of the ball is described.The surface area of**a cube**is equal to S = 6*((D/sqrt(3))^2) = 6*(D^2)/3 = 2*(D^2) = 8*(R^2).