Heron's formula

"What if you know only the three sides, not the height?" Heron, who taught in Alexandria, found a way.

Heron's formula, for sides a, b, c:

• first find the semi-perimeter: s = ½(a + b + c);
• then area = √(s(s – a)(s – b)(s – c)).

Check 1: the 3-4-5 triangle. s = 6, area = √(6 × 3 × 2 × 1) = 6. It is right-angled (3² + 4² = 5²), so ½ × 3 × 4 = 6 too. ✓

Check 2: an equilateral triangle of side a. s = 3a/2, area = √(3a/2 × a/2 × a/2 × a/2) = (√3/4)a², the same as ½ × base × height. ✓

Tip: work out s, then s – a, s – b and s – c, before multiplying anything. Look for square factors to pull out of the root.

Two more formulas link a triangle to its circles:

• area = abc/(4R), where R is the circumradius;
• area = r(a + b + c)/2, where r is the inradius (the radius of the circle that fits snugly inside).