# Heron’s formula (area of triangle)

A colleague was talking about a question in an IGCSE paper where a triangle was shown with the lengths of three sides given. The question asked you to find the area.

This is pretty tricky and involves using the Cos Rule and then the formula: A = 0.5ab sin C

Where a and b are side lengths and C is angle between a and b.

We had come up with another more complex method too.

I’ve just stumbled across something called Heron’s formula. For a triangle with sides a, b and c, define s to be half the perimeter. Then:

Area = sqroot [ s (s-a) (s-b) (s-c) ]

What I love about this is that it’s something we’ve never come across before (despite being mentioned in a book around 60AD) and is a niche application that someone’s taken the effort to work out. I really like the fact that I don’t know everything about maths and keep coming across little gems like Heron’s formula.

Of course, as a mathsy type, I’m wondering if I could prove Heron’s formula and failing that, can I understand a proof? Feels like a future blog post…

I’m a huge fan of Heron’s formula. It can also be used to show that an equilateral triangle with side L has Area = L^2 * sqrt(3)/4. And that tells us the ratio of the area of a square to an equilateral triangle with same side length!