About Heron's Formula
What Is Heron's Formula?
Heron's formula (also called Hero's formula) calculates the area of a triangle when only the three side lengths are known. Named after Hero of Alexandria, it states: A = sqrt(s(s-a)(s-b)(s-c)), where s is the semi-perimeter (a+b+c)/2 and a, b, c are the side lengths.
The Formula
A = sqrt(s(s-a)(s-b)(s-c)) where s = (a+b+c)/2. This formula is remarkable because it does not require knowledge of any angles or heights, only the three sides. It works for any valid triangle.
Related Properties
From Heron's formula, we can derive the inradius r = A/s and the circumradius R = abc/(4A). These additional properties make Heron's formula a powerful tool that unlocks the full geometry of a triangle from just three side measurements.
Applications
Heron's formula is widely used in surveying, navigation, construction, and computer graphics. It is particularly useful when measuring heights or angles is impractical but side lengths can be measured directly.