# Heron's Formula

**Heron's Formula** (sometimes called Hero's formula) is a formula for finding the area of a triangle given only the three side lengths.

## Contents

## Theorem

For any triangle with side lengths , the area can be found using the following formula:

where the semi-perimeter .

## Proof

## Isosceles Triangle Simplification

for all triangles

for all isosceles triangles

simplifies to

## Example

Let's say that you have a right triangle with the sides 3,4, and 5. Your semi- perimeter would be 6. Then you have 6-3=3, 6-4=2, 6-5=1. 1+2+3= 6 The square root of 36 is 6. The area of your triangle is 6.

## See Also

## External Links

In general, it is a good advice **not** to use Heron's formula in computer programs whenever we can avoid it. For example, whenever vertex coordinates are known, vector product is a much better alternative. Main reasons:

- Computing the square root is much slower than multiplication.
- For triangles with area close to zero Heron's formula computed using floating point variables suffers from precision problems.