Area of a Triangle — The General Formula

The familiar area rule, , needs the height — and in a triangle that is not right-angled the height is rarely given. The general formula avoids it completely: two sides and the angle between them are enough.

Theorem Area from two sides and their angle

a and b: the two known sides.

C: the angle between them — the included angle.

Works for any triangle: acute, right or obtuse.

The height is hidden inside the sine: , so becomes . Nothing new is being claimed — the same area, written without .

Note The right angle is just a special case

When , and the formula collapses to — exactly , because the two sides are then the base and the height. The general formula contains the old one.

Example Three applications

In triangle ABC:

⟹ Area = 12 cm²

In triangle XYZ:

(obtuse).

⟹ Area ≈ 30.31 cm²

With

and

fixed,

, so the area is

at 45°:

· at 90°:

· at 135°:

⟹ 45° and 135° give the same area; 90° gives the largest

Reference How the angle changes the area (a = 5, b = 4)

Angle C	sin C	Area
0°	0	0
30°	0.5	5
90°	1 (largest)	10 (maximum)
150°	0.5	5
180°	0	0

The area grows from 0° up to 90°, then shrinks again toward 180°. Supplementary angles give equal areas because .

Summary

Area = ½ × a × b × sin C, where C is the angle included between sides a and b.
It works for every triangle — no height needed.
The area is largest at C = 90°, because sin 90° = 1.
Supplementary included angles (θ and 180° − θ) give the same area.