Hero's formula, also called Heron's formula, relates the area of a triangle to the measures of its three sides. Allowing a, b, and c to denote the lengths of the sides of a triangle, s, the semi-perimeter of the triangle, becomes s = (a + b + c)/2. Hero's formula states simply that the area of a triangle, A, can be expressed as A = sqrt [s(s - a)(s - b)(s - c)]. As a result, the area of the triangle can be determined without knowing its perpendicular height, also known as the triangle's altitude.

Accredited to the Greek geometer Heron (c. 62 AD ), the formula is derived in Heron's most important manuscript, Metrica. Heron's work was discovered in fragmentary form in 1894 and recovered fully in 1896. Recent scholarship, however, suggests that Archimedes (c. 212 BC) may have previously derived the formula.

Although the formula itself is deceptively simple, Heron's...