Triangle Area Calculator (Trigonometry)
Compute the area of any triangle when you know two sides and the angle between them. Ideal for geometry, surveying, and engineering problems where the height is unknown.
About this calculator
When the height of a triangle is not directly available, you can still find the area if you know two sides and the angle they form — called the included angle. The formula is: Area = 0.5 × a × b × sin(C), where a and b are the two known side lengths and C is the included angle between them. This works because the height of the triangle can be expressed as b × sin(C), turning the classic base × height / 2 formula into one that uses the angle directly. The sine function converts the angle into the effective perpendicular height. Angles can be supplied in degrees or radians; if using degrees, they are converted to radians before applying sin().
How to use
Suppose Side A = 8 units, Side B = 5 units, and the included angle = 30°. First, convert 30° to radians: 30 × π/180 ≈ 0.5236 rad. Then apply the formula: Area = 0.5 × 8 × 5 × sin(0.5236) = 0.5 × 8 × 5 × 0.5 = 10 square units. Enter 8 for Side A, 5 for Side B, 30 for the included angle, and select 'degrees' as the angle unit. The calculator returns 10 square units instantly.
Frequently asked questions
What is the formula for triangle area using two sides and an included angle?
The formula is Area = 0.5 × a × b × sin(C), where a and b are the lengths of two sides and C is the angle between them. This is derived from the standard area formula (base × height / 2) by substituting height = b × sin(C). It is part of the SAS (Side-Angle-Side) family of triangle solutions. The result is the same regardless of which two sides and included angle you choose, as long as they are consistently matched.
When should I use the trigonometric formula instead of the base-height formula for triangle area?
Use the trigonometric formula when the perpendicular height of the triangle is not known but two side lengths and their included angle are. This is common in surveying, navigation, and engineering where direct measurement of height is impractical. The SAS approach is also useful in coordinate geometry and vector problems. It avoids the need to construct or measure an altitude explicitly.
Does the calculator work if the angle is given in radians instead of degrees?
Yes. You can select radians as the angle unit, and the calculator passes the value directly to the sine function without any conversion. When degrees are selected, the calculator automatically multiplies the angle by π/180 before computing sin(). The final area result is identical regardless of which unit you choose, as long as you select the correct one. Always double-check your unit setting to avoid errors.