The surface area of a cube is 2400 cm². The length of its diagonal is ______.

Concept

For a cube of edge length a, two formulas matter: the total surface area is S = 6a2 (six identical square faces), and the space diagonal (corner to opposite corner) is d = a√3. So the diagonal is found by first recovering the edge from the surface area, then scaling that edge by √3.

Application

1. Surface area gives the edge: 6a2 = 2400, so a2 = 400 and a = 20 cm.

2. Space diagonal uses d = a√3: d = 20 × √3 = 20√3 cm.

Cross-check

Numerically 20√3 ≈ 20 × 1.732 = 34.64 cm, which is longer than the edge (20 cm) and longer than a face diagonal (a√2 = 20√2 ≈ 28.28 cm) — exactly as a corner-to-corner diagonal of a cube must be, since it spans all three dimensions.