Find the total surface area of a cube if its diagonal is 54 cm.

Let’s build the VD diagonal at the base of the cube.

Let the length of the edge of the cube be equal to X cm. The AED triangle is rectangular and isosceles, then VD ^ 2 = 2 * AB ^ 2 = 2 * X ^ 2.

VD = X * √2 cm.

In a right-angled triangle DBV1, according to the Pythagorean theorem, V1D ^ 2 = VD ^ 2 + BB1 ^ 2.

2916 = 2 * X ^ 2 + X ^ 2 = 3 * X ^ 2.

X ^ 2 = 2916/3 = 972.

X = 18 * √3 cm.

Determine the area of the side face of the cube.

Sgr = X ^ 2 = 972 cm2.

Then the total surface area of the cube is equal to: Sпов = 6 * Sгр = 5832 cm2.

Answer: The total surface area is 5832 cm2.