The Area of a Triangle

Two sides and the angle between them

The familiar area rule, \tfrac12 \times \text{base} \times \text{height}, needs the height — the perpendicular distance from the base to the opposite corner. But often you don't know the height; instead you know two sides and the included angle — the angle that sits between those two sides. Trigonometry turns that straight into an area:

\text{Area} = \tfrac12\,ab\sin C

Here a and b are the two sides and C is the angle between them. It works because the height of the triangle, dropped from the tip of side a, is exactly a\sin C. So \tfrac12 \times \text{base } b \times \text{height } a\sin C = \tfrac12\,ab\sin C.

For a triangle with two sides a and b and the angle C between them:

the area is \tfrac12\,ab\sin C;
C must be the angle between the two sides — the included angle — not just any angle of the triangle;
when C = 90^\circ, \sin 90^\circ = 1, so it collapses to the ordinary \tfrac12 \times \text{base} \times \text{height}.

Seeing the height

Step through the figure: first the two sides a and b with the included angle C, then the dashed height a\sin C that turns it into a half-base-times-height area.