Triangle Area Calculator
Heron's formula finds a triangle's area from its three side lengths alone — no angle or height measurement needed, which is exactly the situation land surveying, carpentry, and geometry homework problems often hand you.
Inputs
Result
14.6969
Perimeter 18
How it works
A = √(s(s−a)(s−b)(s−c))
How the triangle area calculator works
Semi-perimeter: s = (a + b + c) ÷ 2.
Heron's formula: Area = √(s(s−a)(s−b)(s−c)).
For this to produce a real, positive area, the three sides must satisfy the triangle inequality — each side shorter than the sum of the other two.
Worked example: sides 5, 6, 7
- s = (5+6+7) ÷ 2 = 9.
- Area = √(9×(9−5)×(9−6)×(9−7)) = √(9×4×3×2) = √216 ≈ 14.70.
Common mistakes to avoid
Entering three sides that can't form a real triangle
If any one side is longer than or equal to the sum of the other two, the expression under the square root becomes zero or negative — there's no such triangle, not a computational glitch.
Using this when you already know base and height
If a perpendicular height is already known, ½ × base × height is faster and simpler — Heron's formula is specifically for when only the three side lengths are available.
Frequently asked questions
What is the triangle inequality, exactly?
The rule that the sum of any two sides of a triangle must exceed the third side — it's what makes a valid, closeable triangle geometrically possible.
Does this work for any triangle, including obtuse ones?
Yes — Heron's formula works for any valid triangle shape (acute, right, or obtuse) as long as the three side lengths are geometrically valid.
How is the semi-perimeter useful beyond this formula?
It also appears in the formula for a triangle's inradius (the radius of its inscribed circle), making it a genuinely reusable intermediate value in triangle geometry.
What if I know two sides and the angle between them instead?
That calls for a different formula, Area = ½ × a × b × sin(C) — Heron's formula specifically needs all three side lengths, not an angle.
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