# How do you use Heron's formula to find the area of a triangle with sides of lengths 7 , 4 , and 7 ?

Jan 23, 2016

${\text{Area}}_{\triangle} = 6 \sqrt{5}$

#### Explanation:

Heron's formula says that for a triangle with sides of length $a , b , c$ and semi-perimeter $s = \frac{a + b + c}{2}$

${\text{Area}}_{\triangle} = \sqrt{s \left(s - a\right) \left(s - b\right) \left(s - c\right)}$

For the specified triangle
$\textcolor{w h i t e}{\text{XXX}} a = 7$
$\textcolor{w h i t e}{\text{XXX}} b = 4$
$\textcolor{w h i t e}{\text{XXX}} c = 7$
$\rightarrow \textcolor{w h i t e}{\text{XXX}} s = 9$

${\text{Area}}_{\triangle} = \sqrt{9 \left(2\right) \left(5\right) \left(2\right)} = \sqrt{{3}^{3} \times {2}^{2} \times 5} = 6 \sqrt{5}$