Heron's Formula

Find the area of any triangle from its three sides — no height needed

Reviewed by [email protected], Geometry Calculator Developer & Online Math Educator Last updated May 12, 2026

When you know all three sides of a triangle but no angles or heights, Heron's formula gives the area directly. Compute the semi-perimeter s, then plug into a single square root. It works for ANY triangle — scalene, isosceles, equilateral, acute, right, or obtuse.

The Formulas

Name	Formula	Notes
Semi-perimeter	`s = (a + b + c) / 2`	Half the perimeter. Compute first, then plug into the area formula.
Heron's Formula	`A = √[s(s − a)(s − b)(s − c)]`	a, b, c are the three side lengths. The classic form (Hero of Alexandria, ~60 AD).
Algebraic Form	`A = ¼ × √[(a+b+c)(−a+b+c)(a−b+c)(a+b−c)]`	Equivalent expansion — no semi-perimeter step.
Numerically Stable Form	`A = ¼ × √[(a+(b+c))(c−(a−b))(c+(a−b))(a+(b−c))]`	For very thin triangles where standard form loses precision (sort sides a ≥ b ≥ c first).
Triangle Inequality Check	`a + b > c, a + c > b, b + c > a`	All three must hold; otherwise no triangle exists and the radicand goes negative.
Equilateral Special Case	`A = (√3 / 4) × a²`	When a = b = c. Derives from Heron's: s = 3a/2 → A = √[(3a/2)(a/2)³] = √3·a²/4.

Worked Examples

Example 1: Triangle with sides 5, 6, 7

s = (5 + 6 + 7) / 2 = 9
s − a = 9 − 5 = 4; s − b = 9 − 6 = 3; s − c = 9 − 7 = 2
A = √(9 × 4 × 3 × 2) = √216 ≈ 14.697 unit²

Example 2: Right triangle 3-4-5 (verify against ½·b·h)

s = (3 + 4 + 5) / 2 = 6
A = √[6 × 3 × 2 × 1] = √36 = 6
Check via legs: ½ × 3 × 4 = 6 ✓ — Heron agrees.

Example 3: Equilateral triangle side 10

s = 30/2 = 15
A = √[15 × 5 × 5 × 5] = √1875 = 25√3 ≈ 43.30
Check: (√3/4) × 100 = 25√3 ≈ 43.30 ✓

Frequently Asked Questions

What is Heron's formula?

Heron's formula calculates a triangle's area from its three side lengths a, b, c without needing any angle or height. First compute the semi-perimeter s = (a+b+c)/2, then A = √[s(s−a)(s−b)(s−c)].

When should I use Heron's formula?

Use it when you know all three sides (SSS case) but no height or angle. If you also know an angle, the ½·a·b·sin(C) formula is faster. If you know base and height, just use A = ½·b·h.

Does Heron's formula work for right triangles?

Yes — it works for any triangle. For a 3-4-5 right triangle: s = 6, A = √[6·3·2·1] = √36 = 6, which matches ½·3·4 = 6.

What if I get a negative number under the square root?

That means your three sides cannot form a real triangle. Check the triangle inequality: each side must be less than the sum of the other two (a + b > c, etc.).

Who invented Heron's formula?

Heron of Alexandria proved it around 60 AD in his book Metrica. Archimedes likely knew it earlier; modern proofs use coordinates or the Law of Cosines.