# A triangle has sides with lengths: 14, 9, and 12. How do you find the area of the triangle using Heron's formula?

Jan 7, 2016

The area of the triangle is 53.5 square units.

#### Explanation:

Heron's formula is $A = \sqrt{s \left(s - a\right) \left(s - b\right) \left(s - c\right)}$, where $A$ is area, $a , b ,$ and $c$ are the sides of the triangle, and $s$ is the semiperimeter, which is half the perimeter.

The formula for the semiperimeter is $s = \frac{a + b + c}{2}$.

Let side $a = 14$, side $b = 9$, and side $c = 12$.

Calculate the semiperimeter.

$s = \frac{14 + 9 + 12}{2}$

$s = \frac{35}{2} = 17.5$

Calculate the area of the triangle.

$A = \sqrt{s \left(s - a\right) \left(s - b\right) \left(s - c\right)}$

$A = \sqrt{\left(17.5\right) \left(17.5 - 14\right) \left(17.5 - 9\right) \left(17.5 - 12\right)}$

$A = 53.5$ square units