# How do you use Heron's formula to find the area of a triangle with sides of lengths 7 , 5 , and 6 ?

Jan 30, 2016

$A = \sqrt{s \left(s - a\right) \left(s - b\right) \left(s - c\right)} = 6 \sqrt{6} \approx 14.6969$

#### Explanation:

Heron's formula tells us that the area $A$ of a triangle with sides of length $a$, $b$ and $c$ is given by the formula:

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

Where $s = \frac{a + b + c}{2}$ is the semi-perimeter

In our case, let $a = 7$, $b = 5$ and $c = 6$.

Then $s = \frac{a + b + c}{2} = \frac{7 + 5 + 6}{2} = 9$ and we find:

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

$= \sqrt{9 \cdot \left(9 - 7\right) \cdot \left(9 - 5\right) \cdot \left(9 - 6\right)}$

$= \sqrt{9 \cdot 2 \cdot 4 \cdot 3}$

$= \sqrt{36 \cdot 6} = \sqrt{36} \cdot \sqrt{6} = 6 \sqrt{6} \approx 14.6969$