If the diagonal of a cube is 8√3 cm then its surface area is :

For a cube with edge length a, the space diagonal — the segment joining two opposite vertices through the interior — equals a√3. This follows from applying the Pythagorean theorem twice: first to a face diagonal (a√2), then to the right triangle formed by that face diagonal and the vertical edge. The total surface area of a cube, made up of six identical square faces, is S = 6a2.

1. The given space diagonal is 8√3 cm, so equate it to the general formula: a√3 = 8√3.

2. Divide both sides by √3 to isolate the edge length: a = 8 cm.

3. Substitute a = 8 into the surface-area formula: S = 6a2 = 6 × 82.

4. Evaluate the product: S = 6 × 64 = 384 cm2.

Substituting a = 8 cm back into the diagonal formula gives a√3 = 8√3 cm, which reproduces the diagonal stated in the question — confirming that the edge length of 8 cm, and the resulting surface area of 384 cm2, are consistent.