Brahmagupta's formula and squaring a rectangle

"Can a 4-gon's area be found from its four sides?" No. Rhombuses with all sides 3 can have area 9 (a square), 8.01 or 5.41, depending on how much they lean. We need one more fact.

One extra fact that works: the 4-gon is cyclic (its corners lie on a circle). Then Brahmagupta's formula (628 CE) gives:

• s = ½(a + b + c + d);
• area = √((s – a)(s – b)(s – c)(s – d)).

Check: a rectangle a by b has s = a + b, and the formula gives √(b × a × b × a) = ab ✓.

General and special. Adding a condition to a general result gives a special case; the general result is a generalisation. A square is a special case of a rectangle; (a + b)² is (a + b + c)² with c = 0.

Heron is hiding inside Brahmagupta. A triangle is a cyclic 4-gon with one side of length 0. Put d = 0 in Brahmagupta's formula and Heron's formula appears. So Brahmagupta's formula generalises Heron's.

Squaring a rectangle. To 'square' a shape is to build a square with the same area. Baudhāyana's Śhulbasūtra squares a rectangle a by b with a construction that uses ((a + b)/2)² – ((a – b)/2)² = ab.