Heron's Formula: Area of a Triangle from Three Sides

Calculates the area of a triangle from its three sides with Heron's formula √(s(s−a)(s−b)(s−c)), where s is the semi-perimeter.

Explanation

Heron's formula gives the area of a triangle from its three sides alone. No angle, no height. It earns its keep in surveying and anywhere else that lengths are all you can measure.

S = \sqrt{s(s-a)(s-b)(s-c)}, \quad s = \dfrac{a+b+c}{2}

$a$ , $b$ , $c$ — the three side lengths
$s$ — the semi-perimeter, half the distance around the triangle
$S$ — the area

Work out $s$ first, subtract each side from it in turn, multiply the three results together with $s$ , and take the square root.

Example

Take $a = 3$ , $b = 4$ , $c = 5$ . The semi-perimeter is $s = \dfrac{3 + 4 + 5}{2} = 6$ , so

S = \sqrt{6 \times (6-3) \times (6-4) \times (6-5)} = \sqrt{6 \times 3 \times 2 \times 1} = \sqrt{36} = 6

The area is 6. As 3, 4, 5 is a right triangle, this agrees with $\dfrac{1}{2} \times 3 \times 4 = 6$ .

Notes

The sides must satisfy the triangle inequality: any two of them must add up to more than the third. Lengths that fail it describe no triangle, and the calculator reports it.
When the inequality holds, $s - a$ , $s - b$ and $s - c$ all come out positive. For instance $s - a = \dfrac{b + c - a}{2}$ , which is positive exactly when $b + c > a$ . That is why the quantity under the square root is positive and the area is a real number.
Keep the units consistent: sides in cm give an area in cm², sides in m give m².
Three sides determine a triangle completely, so the area they give is unique.