What is Heron's Formula?
Heron's formula (also called Hero's formula) is a method to find the area of a triangle when you know the lengths of all three sides. It is named after Hero of Alexandria, a Greek mathematician who lived around 60 CE.
Why Heron's Formula?
- The standard formula (Area = ½ × base × height) requires knowing the height of the triangle
- For many triangles (especially scalene triangles), finding the height is difficult
- Heron's formula uses only the three side lengths — no height needed!
Heron's Formula:
For a triangle with sides a, b, and c:
- First calculate the semi-perimeter (half the perimeter):
\[
s = \frac{a + b + c}{2}
\]
- Then the area is:
\[
\text{Area} = \sqrt{s(s-a)(s-b)(s-c)}
\]
When to Use Heron's Formula:
- When you know all three sides of a triangle
- When you don't know the height
- Especially useful for scalene triangles (all sides different)
- Works for all types of triangles: acute, obtuse, right, isosceles, equilateral
Example Calculation:
Triangle with sides 5 cm, 12 cm, 13 cm (right triangle):
- s = (5 + 12 + 13)/2 = 30/2 = 15 cm
- Area = √[15(15-5)(15-12)(15-13)] = √[15 × 10 × 3 × 2] = √900 = 30 cm²