How do you use Heron's formula to find the area of a triangle with sides of lengths 12 , 4 , and 7 ?

The Heron formula is

$A = \sqrt{s \left(s - a\right) \left(s - b\right) \left(s - c\right)}$ where $a , b , c$ the sides of the triangle and

$s = \frac{a + b + c}{2}$ the semiperimeter.

Now just replace the values in the formula.

May 26, 2018

There are better choices than Heron, but all lead to an imaginary area because these are not side lengths of a real triangle because they do not satisfy the triangle inequality.

Explanation:

Don't, use one of these instead:

$16 {A}^{2} = 4 {a}^{2} {b}^{2} - {\left({c}^{2} - {a}^{2} - {b}^{2}\right)}^{2}$

$= {\left({a}^{2} + {b}^{2} + {c}^{2}\right)}^{2} - 2 \left({a}^{4} + {b}^{4} + {c}^{4}\right)$

$= \left(a + b + c\right) \left(- a + b + c\right) \left(a - b + c\right) \left(a + b - c\right)$

$16 {A}^{2} = \left(12 + 4 + 7\right) \left(- 12 + 4 + 7\right) \left(12 - 4 + 7\right) \left(12 + 4 - 7\right) = 23 \left(- 1\right) \left(15\right) \left(9\right)$

The negative squared area shows this triangle doesn't satisfy the triangle inequality.