The perimeter of a rhombus whose diagonals are 10 cm and 24 cm is

Concept: In any rhombus, the two diagonals bisect each other at right angles (90°). This splits the rhombus into four congruent right triangles, each with legs equal to half of each diagonal and a hypotenuse equal to the rhombus’s side. By the Pythagorean theorem, side = √(half-diagonal2 + half-diagonal2).

Application

1. Halve each diagonal: 10 ÷ 2 = 5 cm and 24 ÷ 2 = 12 cm — these are the legs of each right triangle formed at the centre.

2. Apply the Pythagorean theorem to get the rhombus’s side: side = √(52 + 122) = √(25 + 144) = √169 = 13 cm.

3. A rhombus has four equal sides, so perimeter = 4 × side = 4 × 13 = 52 cm.

Cross-check

5, 12, 13 is a well-known Pythagorean triple, confirming the side is exactly 13 cm (not an approximation). Equivalently, side = ½√(d12 + d22) = ½√(102 + 242) = ½√(100 + 576) = ½√676 = ½ × 26 = 13 cm — the same result.

So the perimeter of the rhombus is 52 cm.