GeoProof — Triangle Area ½·a·b·sin C explorer

Area = ½·a·b·sin C

Area from Two Sides and the Angle

If you know two sides of a triangle and the angle between them, the area follows at once:

Area = ½·a·b·sin C

Here a and b are the two sides meeting at C, and C is the included angle between them.

Why it works

Take side a as the base. The height to the opposite vertex is the other side b tipped by the angle: h = b·sin C. Plug into ½·base·height and you get ½·a·b·sin C. Toggle the height on to see it.

The right-angle case

When C = 90°, sin C = 1 and the formula becomes ½·a·b — the familiar "half base times height" for a right triangle, since the two sides are already perpendicular.

A useful relative

Combined with the law of sines, this leads to Area = abc / (4R), linking a triangle's sides to the radius R of its circumscribed circle.