Heron’s formula
With sides a, b, c and half-perimeter s = (a + b + c)/2, the area is √[s(s − a)(s − b)(s − c)].
09 / ANTIQUITY TO THE RENAISSANCE
Heron of Alexandria
A surveyor can walk around a field more easily than measure straight across it. Heron’s formula turns the three boundary lengths into an area.
Explore the mathematics ↓
01 / TRIANGLE AREA
How much land is inside three fences?
Walking the fences of a field is easy. Measuring straight across it, at a right angle to one side, is hard.
Heron gave a recipe for the area of a triangle using only its three side lengths.
Can fence lengths tell you how much land is inside?
This field has four fences: 6, 4, 6 and 4 metres. Drag its corner. The fences never change length.
Fence lengths
=Land inside
=Land inside
Where does Heron’s recipe come from?
Call the base c and the other fences a and b. A height h meets the base x from one end, so b² = x² + h² and a² = (c − x)² + h².
Subtracting gives x, then area² = c²h²/4. After careful algebra this becomes s(s − a)(s − b)(s − c), with s half the perimeter. Heron’s Metrica gives a geometric proof.
THE IDEAFour fences can wobble; three cannot. So three fence lengths fix a triangle’s area, and Heron’s recipe computes it: √(s(s − a)(s − b)(s − c)), where s is half the perimeter.
The person behind the ideas
- A mathematician and engineer working in Roman-era Alexandria.
- His Metrica collects methods for finding areas and volumes.
- His writings also describe mechanical devices and experiments with air and steam.
More of the mathematics
Practical mechanics
Heron wrote about surveying instruments, automata, and devices powered by air, water, and steam.