Heron's Formula

With three sides and no angle, there is a formula with no trig at all. Let s be half the perimeter, $s = \frac{a+b+c}{2}$. Then the area is:

$$A = \sqrt{s\left(s-a\right)\left(s-b\right)\left(s-c\right)}$$

For sides 5, 6 and 7: s = 18/2 = 9, and $A = \sqrt{9\cdot 4\cdot 3\cdot 2}$, which is √216 = 6√6, about 14.70. The other way round gives the same: the law of cosines for one angle, then the area with its sine. Heron skips the angle.

On the board, the formula is written for your triangle with s still a letter. Work out s first, put it into A, then simplify and round.