Heron's formula

How to use it

1. 1Use it only when the three sides are known and no height is: with a base and its height, $\frac{1}{2}bh$ is far quicker.
2. 2Compute $s$ first and keep the four factors as exact fractions; halves cancel in pairs and the surd usually simplifies.
3. 3It applies to any triangle whatever, so it is the fallback for an awkward piece of a compound figure.

Step-by-step

1. 1
   $s=\frac{5+6+7}{2}=9$
2. 2
   $A=\sqrt{9\times 4\times 3\times 2}=\sqrt{216}=6\sqrt{6}$

   Why: The four factors are $s$, $s-a$, $s-b$ and $s-c$.