Triangle Area from 3 Sides (Heron's Formula)
Enter the three side lengths of any triangle and get its area by Heron’s formula, plus the semi-perimeter and perimeter.
How it works
s = (a + b + c) ÷ 2 · Area = √( s·(s − a)·(s − b)·(s − c) )
Heron’s formula finds a triangle’s area from its three side lengths alone, with no need to know any angle or height. First compute the semi-perimeter s, half the total of the sides, then take the square root of the product s(s − a)(s − b)(s − c). It only produces a real area when the three sides actually satisfy the triangle inequality — otherwise the sides cannot close into a triangle.
Worked example
For sides 5, 6 and 7: s = (5 + 6 + 7) ÷ 2 = 9. Area = √(9·(9−5)·(9−6)·(9−7)) = √(9·4·3·2) = √216 = 14.70 square units.
Sources: Heron's formula — Wikipedia
Frequently asked questions
When is Heron’s formula the right tool?
Use it whenever you know all three side lengths but no angle or height. If you know a base and its perpendicular height, the simpler ½·base·height is faster.
What is the triangle inequality?
For three lengths to form a triangle, each one must be less than the sum of the other two. If that fails — say sides 1, 2 and 10 — the sides cannot meet, and this calculator flags it instead of returning a bogus area.
Does it work for right triangles?
Yes. Heron’s formula works for every valid triangle. For a 3-4-5 right triangle it gives area 6, matching ½·3·4.
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This calculator is for educational purposes. Double-check important results before acting on them.