# Heron’s Formula Calculator

Enter the three sides of a triangle to calculate the area using Heron’s formula.

## Solution:

## How to use Heron’s Formula

Using Heron’s formula is an easy way to calculate the area of a triangle given the length of the sides. The most commonly used formula for the area of a triangle requires knowing the height of the triangle, which often requires knowing the interior angles to determine.

However, Heronâ€™s formula does not require knowing the height of the triangle. Thus, the interior angles of the triangle are not needed to find area using this formula.

For a triangle with three side lengths *a*, *b*, and *c*, and the semiperimeter *s*, Heron’s formula can be used to find the area.

A = s(s – a)(s – b)(s – c)

Thus, the area *A* of a triangle is equal to the square root of the semiperimeter *s* times *s* minus side *a* times *s* minus side *b* times *s* minus side *c* .

The semiperimeter *s* in Heron’s formula is half the perimeter, so the equation to find *s* is:

s = a + b + c / 2

The semiperimeter *s* is equal to the quantity side a plus side b plus side c, divided by 2.

After solving the semiperimeter, substitute *s* in Heron’s formula above, along with side lengths *a*, *b*, and *c*, to solve for the area.

### Alternate Heron’s Formula Equations

Heron’s formula can be simplified or rewritten in a few different ways to solve for area. The following formulas are all derived from Heron’s formula with respect to side lengths *a*, *b*, and *c*.

A = 1 / 4(a + b + c)(-a + b + c)(a – b + c)(a + b – c)

A = 1 / 42(a²b² + a²c² + b²c²) – (a⁴ + b⁴ + c⁴)

A = 1 / 4(a² + b² + c²)² – 2(a⁴ + b⁴ + c⁴)

A = 1 / 44(a²b² + a²c² + b²c²) – (a² + b² + c²)²

A = 1 / 44a²b² – (a² + b² – c²)²