Topics · 2 identities

Area

SAS and Heron's formulas for the area of a triangle.

Triangle area from two sides and the included angle: ½ab sin C

\text{Area} = \tfrac{1}{2} a b \sin C

Area = ½ab sin C gives a triangle's area from two sides and the angle between them. It is the familiar ½ × base × height with b sin C standing in for the height. The angle must be the included one - the one between the two sides you used.

Open identity

Heron's formula: triangle area from three sides

\text{Area} = \sqrt{s(s-a)(s-b)(s-c)},\ \ s = \tfrac{a+b+c}{2}

Heron's formula gives the area from the three sides alone: √(s(s−a)(s−b)(s−c)), where s = (a + b + c)/2 is the semiperimeter. No angle is needed, which makes it the one to use for SSS. If any factor comes out negative, the three lengths cannot form a triangle.

Open identity