How do I find a triangle's area with two sides and an angle?
Use Area = ½·a·b·sin C, where a and b are two sides and C is the angle between them. It works for any triangle and needs no perpendicular height.
// maths › Non-Right-Angle Trigonometry
Area of any triangle (½·a·b·sin C)
Find the area of any triangle from two sides and their included angle using one half a b sine C — the area companion to the sine and cosine rules — in degrees or radians, with the triangle drawn in the standard a, b, c labelling and the area shaded.
Area = ½·a·b·sin C (a, b sides; C the angle between them)
Angle unit
Frequently asked questions
How is this different from the basic area formula?
The basic ½ × base × height needs the perpendicular height. This version replaces height with b·sin C, so you can find the area straight from two sides and their included angle — handy alongside the sine and cosine rules.
Can you give a worked example?
With a = 8, b = 11, and the included angle C = 37°: Area = ½·8·11·sin 37° ≈ 44·0.602 ≈ 26.5 square units. The same angle in radians is 37° ≈ 0.6458 rad; flip the Degrees/Radians toggle and the working updates to match.
Which angle do I use?
Always the angle enclosed by the two sides you chose. With standard labelling, sides a and b enclose angle C; if you use sides b and c, the included angle is A instead.
Where is this used in real life?
Surveyors and land valuers find areas of irregular plots from measured sides and angles, architects compute triangular surfaces, and it is used in engineering cross-sections and 3D mesh modelling.