Area of Any Triangle

The familiar ½ × base × height needs a height, and a slanted triangle rarely hands you one.

The sine formula

Area = ½ab sin C, where a and b are two sides and C is the angle between them.

Where it comes from

The height dropped from one vertex is b sin C. Substituting that into ½ × base × height gives the formula directly.

The angle must be between

C has to be the included angle. Any other angle gives the wrong height and the wrong area.

When you have three sides

With no angle at all, use Heron’s formula. Let s = (a + b + c) ÷ 2, then area = √(s(s−a)(s−b)(s−c)).

Area when you cannot see the height

Area = ½ab sin C uses two sides and the angle between them. It comes from ½ × base × height, with the height written as a sin C — trigonometry supplying the height you cannot measure directly.

The angle must be included

The angle in the formula must be the one between the two sides used. Substituting a non-included angle gives an answer that is wrong without looking wrong, which makes it a dangerous error.

Heron's formula for three sides

With all three sides and no angle, Heron's formula gives the area from the semi-perimeter. It needs no trigonometry at all and is the right tool when the SSS case arises.

Where it is used

Surveying irregular land, computing forces on triangular structures, and any graphics application that shades triangles. Triangles are the universal building block of computational geometry, and their area is computed constantly.