Triangle Area (two sides & angle)

Know two sides of a triangle and the angle between them? This calculator applies the SAS area formula, A = ½·a·b·sin C, and returns the area in square units.

How the calculation works

The area of a triangle equals half the product of two sides times the sine of the included angle: A = ½ × a × b × sin C. It comes straight from A = ½ × base × height, because the height dropped onto side a is b × sin C.

Enter side a, side b (the default unit is meters, but any length unit works) and the included angle C in degrees, between 0 and 180°. “Included” matters: the angle must sit between the two sides you entered. The area comes out in the square of your length unit — feet give square feet, centimeters give square centimeters.

The tool converts degrees to radians internally and shows the formula used, ½ × a × b × sin(γ), so you can check the working or copy it into homework.

Worked example

Defaults: a = 5 m, b = 7 m, C = 60°. sin 60° = √3/2 ≈ 0.8660, so A = 0.5 × 5 × 7 × 0.8660 ≈ 15.16 m². Change the angle to 90° and the area rises to its maximum of 17.5 m²; at 150° it drops to 8.75 m², exactly the same as at 30°.

Things to keep in mind

• Supplementary angles have the same sine, so 30° and 150° give the same area for the same two sides.
• For fixed sides the area peaks at 90°, where the formula reduces to ½ab — the right-triangle area.
• At 0° or 180° the triangle collapses into a line and the area is zero.
• If the angle you know is opposite one of the sides rather than between them, use the law of sines first to find the included angle (watch out for the ambiguous SSA case).