Heron's Formula

Area of a Triangle from Sides

We can calculate the area of a triangle if we know the lengths of all three sides, using a formula that has been known for nearly 2000 years.

It is called "Heron's Formula" after Hero of Alexandria (see below)

We use this two step process:

Step 1: Calculate "s" (half of the triangle's perimeter, called the semiperimeter):

s = a+b+c2

Step 2: Then calculate the Area:

Example: What's the area of a triangle with sides 3, 4, and 5?

We already know its area is 3 × 42 = 6

But let's use Heron's formula:

Step 1: s = 3 + 4 + 52 = 122 = 6

Step 2: A = √6(6−3)(6−4)(6−5) = √6 × 3 × 2 × 1 = √36 = 6

It works perfectly!

Try it yourself:

Sanity Check

Does your triangle make sense? Make sure the three side lengths are able to make a real triangle.

The sides must satisfy the triangle inequality theorem:

a + b > c
a + c > b
b + c > a

If not, there's no triangle, the area isn't defined, and the formula won't work.

Hero of Alexandria

The formula is credited to Hero (or Heron) of Alexandria, who was a Greek Engineer and Mathematician in 10 – 70 AD.

Amongst other things, he developed the Aeolipile, the first known steam engine, but it was treated as a toy!

Angles

In the calculator above I also used the Law of Cosines to calculate the angles (for a complete solution). The formula is:

Where "C" is the angle opposite side "c".

But we don't need the angles to use Heron's Formula.