Herons Formula Heron's Formula

The familiar area rule, one-half times base times height, is easy when you know a height. But for a scalene field or a triangular park you often know only the three side lengths and have no easy height to measure. Heron of Alexandria gave a formula that solves exactly this: feed in the three sides and it returns the area directly. It is one of the most useful results in school geometry because real land rarely comes with a height marked on it.

Every use of Heron's formula begins with the semi-perimeter, written s. Add the three sides and divide by two: s = (a + b + c)/2. The word 'semi' means half, so s is half the distance round the triangle. This single number then appears in all four brackets of the formula, so getting it right first makes the rest straightforward.

With s known, the area is the square root of s times (s minus a) times (s minus b) times (s minus c). Each bracket subtracts one side from the semi-perimeter. Multiply the four quantities together, then take the square root. The answer comes out in square units, the same units squared as the sides. Because it uses a square root, it is fine for the answer to be an irrational number such as 6 root 6.

Heron's formula is for triangles, but it unlocks four-sided figures too. Draw a diagonal across a quadrilateral and it becomes two triangles. If the diagonal length and all four sides are known, you can find each triangle's area by Heron's formula and add them. This is how surveyors estimate the area of irregular plots of land that are not neat rectangles.