What is Heron's formula?
Heron's formula gives a triangle's area from its three sides alone: Area = sqrt(s(s-a)(s-b)(s-c)), where s = (a+b+c)/2 is the semi-perimeter. No angle or height is needed.
// maths › Triangles
Heron's formula
Find the area of any triangle from its three side lengths alone using Heron's formula, the square root of s times s minus a times s minus b times s minus c, where s is the semi-perimeter, with the triangle drawn to scale and the area shaded.
Area = √(s(s−a)(s−b)(s−c)), s = (a+b+c)/2
A mind behind this: Hero of Alexandria c. 10 – c. 70 AD
Frequently asked questions
Why is this in Geometry rather than Trigonometry?
Because the formula uses only side lengths, with no angle anywhere in it — that makes it a result of pure geometry. Trigonometry studies the relationship between sides and angles; Heron's formula needs no angle, so its natural home is geometry.
Can you give a worked example?
For a 3-4-5 triangle: s = (3+4+5)/2 = 6, so Area = sqrt(6 x 3 x 2 x 1) = sqrt(36) = 6 square units, matching the familiar right-triangle area.
Who discovered it?
It is credited to Hero (or Heron) of Alexandria, who proved it in his work Metrica around 60 AD, though it may have been known to Archimedes two centuries earlier and was found independently in China by Qin Jiushao in 1247.
Where is this used in real life?
Surveyors and civil engineers find land and plot areas from side measurements, architects and builders work out triangular areas, and it is used in computer graphics to find the area of triangular faces in 3D meshes.