Question

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

Answer 1

The Heron formula is

`A=sqrt[s(s-a)(s-b)(s-c)]` where `a,b,c` the sides of the triangle and

`s=[a+b+c]/2` the semiperimeter.

Now just replace the values in the formula.

Answer 2

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:

`16A^2 = 4a^2b^2-(c^2-a^2-b^2)^2`

`= (a^2+b^2+c^2)^2-2(a^4+b^4+c^4)`

`=(a+b+c)(-a+b+c)(a-b+c)(a+b-c)`

`16A^2 = (12+4+7)(-12+4+7)(12-4+7)(12+4-7) = 23(-1)(15)(9)`

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