How this instrument works
Heron's formula finds a triangle's area from its three side lengths alone, with no angle or height needed anywhere in the calculation. First combine the three lengths into a semi-perimeter, s = (a + b + c) ⁄ 2; then the area follows as the square root of s times each of the three gaps between s and a side, A = √(s(s−a)(s−b)(s−c)). The relation holds for any triangle, scalene or otherwise, since nothing in it assumes a particular shape.
This site's other scalene-triangle page runs a distinctness test first, (a−b)(b−c)(c−a), to confirm that three entered lengths genuinely form a scalene shape with no matching pair, reporting area only as a secondary figure alongside that test. This page skips the classification step entirely and assumes it's already settled: feed in three sides already known to be scalene, and the area comes straight from Heron's formula with nothing else attached.
A useful cross-check turns up for right triangles: when the three sides also happen to satisfy the Pythagorean theorem, Heron's formula and the simpler ½ × base × height both land on the identical figure, since a right triangle is just a scalene triangle with one angle fixed at 90°.
- Enter the three side lengths into Side a, Side b, and Side c, in any consistent unit.
- Read Area — the sheet builds the semi-perimeter first, then applies Heron's formula automatically.
- Keep the three lengths consistent with the Triangle Inequality (any two sides must sum to more than the third), or the result is flagged as invalid.
- Enter the sides in any order — a, b, and c are interchangeable, and the resulting area stays identical regardless of which length goes where.
Worked example — sides 5, 6, and 7
Sides 5, 6, and 7 give a semi-perimeter of s = (5 + 6 + 7) ⁄ 2 = 9. Heron's formula then produces area = √(9 × (9−5) × (9−6) × (9−7)) = √(9 × 4 × 3 × 2) = √216 ≈ 14.696938456699069.
Sides 3, 4, and 5 — the classic right triangle — give s = (3 + 4 + 5) ⁄ 2 = 6, and area = √(6 × 3 × 2 × 1) = √36 = 6.0 exactly, matching the simpler ½ × 3 × 4 = 6 that a right triangle's two legs allow directly. Sides 7, 8, and 9 give s = 12, and area = √(12 × 5 × 4 × 3) = √720 ≈ 26.832815729997478.
Questions
What is Heron's formula for triangle area?
Area = √(s(s−a)(s−b)(s−c)), where s is the semi-perimeter, (a+b+c) ⁄ 2. For sides 5, 6, and 7, s = 9 and the area works out to √216 ≈ 14.697.
Do I need to confirm the sides form a scalene triangle before using this page?
Only if that classification matters to you — this page assumes it's already settled and computes area alone. This site's other scalene-triangle page runs a distinctness test, (a−b)(b−c)(c−a), specifically to confirm no two sides match, before also reporting area.
Does the order I enter a, b, and c matter?
No — the semi-perimeter and the resulting area come out identical no matter which length is labeled a, b, or c, since addition and multiplication don't care about label order. Only the Triangle Inequality among the three raw lengths matters, not which letter each gets assigned.
Why does Heron's formula match ½ × base × height for a right triangle?
Because a right triangle is still a triangle, and Heron's formula works for every triangle regardless of shape. Sides 3, 4, and 5 give area = 6 either way — through Heron's formula, or directly as half the product of the two legs, 3 and 4, since a right triangle's legs double as base and height.
What happens if the three lengths I enter can't form a triangle at all?
One of the terms under the square root turns negative, since the longest length exceeds the sum of the other two. This calculator flags that combination as invalid rather than returning a meaningless figure for lengths that could never close into an actual shape.
Can Heron's formula be used on an isosceles or equilateral triangle too?
Yes — it works for any triangle at all, regardless of whether the three sides are all different, only two match, or all three are equal. The scalene case here is simply one instance of a formula that never assumes anything about which sides, if any, happen to be equal.