# A triangle has sides with lengths: 3, 2, and 4. How do you find the area of the triangle using Heron's formula?

Dec 27, 2015

Area $= \frac{3}{4} \sqrt{15}$

#### Explanation:

Given a triangle with sides $a , b , c$
and semiperimeter $s \left(= \frac{a + b + c}{2}\right)$

Heron's formula tells us that the area is:
color(white)("XXX")"Area"_triangle = sqrt(s(s-a)(s-b)(s-c))

Using the given values $\left(a , b , c\right) = \left(3 , 2 , 4\right)$
$\textcolor{w h i t e}{\text{XXX}} s = \frac{9}{2}$
and
color(white)("XXX")"Area"_triangle = sqrt((9/2)(3/2)(5/2)(1/2))

$\textcolor{w h i t e}{\text{XXX}} = \frac{\sqrt{135}}{4}$

$\textcolor{w h i t e}{\text{XXX}} = \frac{3 \sqrt{15}}{4}$