How to find the area of a triangle using sine?

The sine area formula explained

In geometry, the standard area of a triangle is half of the base multiplied by the perpendicular height. However, you often know two side lengths and the angle between them instead of the height. Solving this problem directly is easy by using trigonometry.

Consider a triangle with side lengths a and b enclosing an angle C. Dropping a perpendicular line from the top vertex creates a right triangle. In that right triangle, the height equals b * sin(C). Substituting this expression into Area = 1/2 * base * height gives the formula Area = 1/2 * a * b * sin(C).

Step-by-step calculation

First, verify that your known angle sits directly between your two known sides. This is called the included angle.

Second, ensure your calculator is set to degree mode if the angle is measured in degrees, or radian mode if it is given in radians.

Third, multiply side a by side b, multiply by the sine of angle C, and divide the result by 2. For instance, if side a is 8 cm, side b is 10 cm, and angle C is 30 degrees, Area = 0.5 * 8 * 10 * sin(30°). Since sin(30°) = 0.5, the area equals 20 square centimeters.

Finding missing values and extended shapes

You can rearrange the formula to find a missing side or angle when the area is already known. For example, sin(C) = (2 * Area) / (a * b). If the area is 15 square meters, side a is 6 meters, and side b is 10 meters, sin(C) = 30 / 60 = 0.5, which gives an angle of 30 degrees or 150 degrees.

This formula also applies to parallelograms. Because a diagonal splits a parallelogram into two congruent triangles, the total area of a parallelogram is simply a * b * sin(C).