GeoProof — Triangle Area = abc / 4R

Area = abc / (4R)

A triangle's area can be written from its three sides and the radius of its circumscribed circle:

Area = abc / (4R)

a, b, c are the side lengths and R is the circumradius — the radius of the unique circle through all three vertices.

Where it comes from

Two area formulas combine: Area = ½·a·b·sin C, and the law of sines gives sin C = c / (2R). Substituting, Area = ½·a·b·(c/2R) = abc/(4R).

Rearranged

Solving for the circumradius gives R = abc / (4·Area) — a quick way to find the circle through three points once you know the area (e.g. from Heron's formula).

A feel for it

For fixed side lengths, a flatter triangle has a smaller area and a larger circumscribed circle; squeeze the area toward zero and R grows without bound as the three points line up.