Scalene Triangle Area Calculator — Heron's Formula
Enter all three side lengths of a scalene triangle (a triangle with no two equal sides) to get the area via Heron's formula, the perimeter, all three heights, and all three interior angles.
Computed via Heron's formula from the three side lengths
- 1
Semi-perimeter s = (a + b + c) ÷ 2
(5 + 7 + 9) ÷ 2 = 10.5 - 2
Product s(s−a)(s−b)(s−c)
10.5 × 5.5 × 3.5 × 1.5 = 303.1875Heron's formula — the area is the square root of this product. - 3
Area = √(product)
√(303.1875) = 17.4123
How does this calculator work?
For a scalene triangle with sides a, b, c: compute s = (a+b+c)/2, then Area = √[s(s−a)(s−b)(s−c)] (Heron's formula). Heights are hₐ = 2·Area/a, hb = 2·Area/b, hc = 2·Area/c. Angles follow from the law of cosines. All sides must satisfy the triangle inequality.
Formula
How this is calculated
A scalene triangle has three sides of distinct lengths — no two sides are equal, so no two angles are equal. Given side lengths a, b and c, the semi-perimeter s = (a + b + c) / 2 is computed first. Heron's formula then gives the area as √[s(s − a)(s − b)(s − c)] without needing a height or angle to be measured directly. It is particularly useful for triangles defined by surveyed distances.
The three altitudes (heights) follow from the area: hₐ = 2 · Area / a, and symmetrically for the other two sides. All three interior angles are derived from the law of cosines: cos A = (b² + c² − a²) / (2bc), and so on. The angles sum to exactly 180°.
The triangle inequality (each side < sum of the other two) must hold or no real triangle exists — the calculator will show a warning if your inputs violate it. All calculations assume a flat Euclidean plane.
Frequently asked questions
Heron's formula computes the area of any triangle from its three side lengths alone, without requiring a height. It states Area = √[s(s−a)(s−b)(s−c)] where s = (a+b+c)/2 is the semi-perimeter. It works for all triangles — scalene, isosceles or equilateral.
A scalene triangle has all three sides of different lengths, which means all three interior angles are also different. It is the most general type of triangle — isosceles (two equal sides) and equilateral (all equal) are special cases.
The triangle inequality requires that each side is strictly shorter than the sum of the other two. If any side is too long — for example a = 10, b = 3, c = 3 — no real triangle can be formed and the area is undefined.
Also known as
TG we-Calculate Editorial Team. (2026). Scalene Triangle Area Calculator — Heron's Formula [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/scalene-triangle-area-calculator
TG we-Calculate Editorial Team. "Scalene Triangle Area Calculator — Heron's Formula." TG we-Calculate. 2026. https://we-calculate.com/calculator/scalene-triangle-area-calculator.
TG we-Calculate Editorial Team, "Scalene Triangle Area Calculator — Heron's Formula," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/scalene-triangle-area-calculator
@misc{wecalculate_scalene_triangle_area_calculator, title = {Scalene Triangle Area Calculator — Heron's Formula}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/scalene-triangle-area-calculator}}, year = {2026}, note = {TG we-Calculate} }
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