Heron's formula

Statement

Let $ABC$ be a triangle in the Euclidean plane, with the side lengths $a$, $b$ and $c$. Let $s = \frac{1}{2}(a + b + c)$ be its semiperimeter. Then the area $T$ of the triangle is $$T = \sqrt{s (s - a)(s - b)(s - c)}.$$

Remarks

An equivalent form without $s$ is $16 T^2 = (a + b + c)(-a + b + c)(a - b + c)(a + b - c)$. For example, the triangle with the sides $4$, $13$ and $15$ has $s = 16$ and $T = \sqrt{16 \cdot 12 \cdot 3 \cdot 1} = 24$. One proof uses the law of cosines to get $\cos \gamma$ from the sides, and then $T = \frac{1}{2} a b \sin \gamma$.

History

The formula carries the name of Heron of Alexandria, who gave a proof in his Metrica in the 1st century. It was possibly known earlier. Qin Jiushao published an equivalent formula in 1247.

Explanations

L1 human

Heron's formula gives the area of a triangle when you know only its three sides. You do not need a height or an angle. For the right triangle with the sides $3$, $4$, $5$: $s = 6$, and $\sqrt{6 \cdot 3 \cdot 2 \cdot 1} = 6$, which is also $\frac{1}{2} \cdot 3 \cdot 4$.