# A triangle has sides with lengths: 4, 12, and 2. How do you find the area of the triangle using Heron's formula?

Jan 16, 2016

No such triangle is possible since no triangle can have a side which is longer than the sum of the other two sides (at least in Euclidean space)

#### Explanation:

If you attempted to apply Heron's formula:
$\textcolor{w h i t e}{\text{XXX}} A r e a = \sqrt{s \left(s - a\right) \left(s - b\right) \left(s - c\right)}$
for a triangle with sides $a = 4 , b = 12 , c = 2$
and semi-perimeter $s = 9$ (i.e. $\frac{a + b + c}{2}$)

you would end up attempting to find the square root of a negative number: $\sqrt{9 \cdot \left(5\right) \cdot \left(- 3\right) \cdot \left(7\right)}$