Heron's Formula (Triangle Area from Three Sides)

Heron's formula computes a triangle's area from its three sides alone — no angle, no height, no trigonometry. Halve the perimeter to get $s$, then $A = \sqrt{s(s-a)(s-b)(s-c)}$. It is named for Hero of Alexandria, who published it around 60 AD, though Archimedes likely knew it three centuries earlier.

Its practical value is that sides are what you can actually measure. A surveyor with a tape but no theodolite, or a builder checking an irregular lot, can get an exact area from three distances. The famous 13-14-15 triangle gives a satisfyingly whole answer: $s = 21$, and $A = \sqrt{21 \times 8 \times 7 \times 6} = \sqrt{7056} = 84$.

The formula also fails informatively. If the three lengths cannot form a triangle, one of the bracketed terms goes negative and the square root is imaginary — the algebra refusing to draw an impossible shape.