# How do you use Heron's formula to find the area of a triangle with sides of lengths 23 , 21 , and 20 ?

Feb 10, 2016

Find the semiperimeter first then use Heron's formula
Area is $24 \sqrt{66}$

#### Explanation:

Heron's formula states that
Area = $\sqrt{s \left(s - a\right) \left(s - b\right) \left(s - c\right)}$

Where s is the sum of all sides / 2 (Also known as the semiperimeter)
a,b,c are the side lengths

From here, you can just plug in all the values.
Find that s = 32
Area = $\sqrt{32 \left(9\right) \left(11\right) \left(12\right)}$
prime factorize the inside to get $\sqrt{\left({2}^{7}\right) \left({3}^{3}\right) \left(11\right)}$

Then take out the squares to get
${2}^{3} \cdot 3$$\sqrt{2 \cdot 3 \cdot 11}$
Finally
$24 \sqrt{66}$