Heron's formula as a corollary

Combining the trigonometric formula for the area with the cosine rule, we arrive at one of the most famous formulae in geometry: Heron’s formula, which expresses the area of a triangle in terms of the sides alone.

Remark — Heron's formula as a corollary

Combining the cosine rule and the area formula we can derive Heron’s formula $S = \sqrt{s(s-a)(s-b)(s-c)}, \qquad s = \frac{a+b+c}{2} \ \text{(semiperimeter).}$ The proof is algebraically laborious but not conceptually difficult: one isolates $\cos\widehat{A}$ from the cosine rule, substitutes it into $\sin^2\widehat{A}=1-\cos^2\widehat{A}$, and expresses $S^2 = \left(\tfrac{1}{2}bc\sin\widehat{A}\right)^2$ in terms of $a,b,c$. We discuss it in the exercises.

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Topics: Triangle trigonometry
Concepts: Triangle area · Heron’s formula · Cosine rule
Skills: Proving