GeoProof — Heron's Formula explorer

a (BC) b (CA) c (AB)

Area = √( s(s−a)(s−b)(s−c) ) where s = semiperimeter = (a+b+c)/2

Heron's Formula

Heron's formula gives the area of a triangle directly from its three side lengths a, b, c — no height or angle needed. With the semiperimeter s = (a + b + c) / 2:

Area = √( s(s−a)(s−b)(s−c) )

The semiperimeter is simply half the perimeter; the three factors (s − a), (s − b), (s − c) are each positive exactly when the sides can form a triangle.

The triangle-inequality link

If one side is too long to close up (say a > b + c), then (s − a) becomes negative and the product under the root goes negative — the formula reports no real area, signalling that no triangle exists. Type a = 10, b = 1, c = 1 to see it fail.

Part of a bigger family

Add a fourth side and Heron's formula generalizes to Brahmagupta's formula for a cyclic quadrilateral, √((s−a)(s−b)(s−c)(s−d)). Heron is the case where one side shrinks to zero.

A 3-4-5 check

For sides 3, 4, 5: s = 6, so Area = √(6·3·2·1) = √36 = 6 — matching ½·3·4 for that right triangle.

A bit of history

It is named for Heron of Alexandria (c. 10–70 CE), who gave a proof in his work Metrica, though the result may have been known earlier to Archimedes.