Heron's Formula

Triangle area from three sides: A = √s(s-a)(s-b)(s-c).

The idea

Theorem (Heron's formula).

A triangle with side lengths $a$, $b$, $c$ has area $A = \sqrt{s(s-a)(s-b)(s-c)}, \qquad s = \frac{a+b+c}{2},$ where $s$, half the perimeter, is called the semiperimeter.

The formula computes the area from the three side lengths alone. The rule $A = \tfrac{1}{2}(\text{base})(\text{height})$ requires a height, and a height is not part of a triangle's data — we construct it by dropping a perpendicular. Three side lengths, however, determine a triangle completely, so they determine its area, and Heron's formula states that dependence.

Each factor $s-a$, $s-b$, $s-c$ is positive exactly when the corresponding side is shorter than the other two combined, which is the condition for the three lengths to form a triangle at all. As one factor approaches $0$ the triangle flattens toward a segment, and the formula sends the area to $0$ with it.

Ways to work on it

• Walkthrough. Find a triangle's area from its three side lengths, step by step.
• Proof. See why — square 12 ab C and bring in the law of cosines.
• Practice. Heron's formula on triangles with clean integer areas.
• Hardest. Irrational area of an equilateral triangle.

Not sure where to start? Take the ten-question placement test.