# How do you use Heron's formula to find the area of a triangle with sides of lengths 12 , 5 , and 7 ?

Jan 21, 2016

The given numbers cannot be lengths of sides of a triangle. (The area is zero)

#### Explanation:

The Heron's formula says that for any triangle with sides $a , b , c$ its area is $A = \sqrt{p \left(p - a\right) \left(p - b\right) \left(p - c\right)}$, where $p = \frac{a + b + c}{2}$

If we substitute given numbers we see, that:

$p = \frac{12 + 5 + 7}{2} = 12$, so

$A = \sqrt{12 \cdot 0 \cdot 7 \cdot 5} = 0$

You should have noticed, that given numbers cannot be the lengths of sides of a triangle, because $12 = 7 + 5$ and according to triangle inequalities each side must be smaller then the sum of remaining sides