The conventional formula for the area of a triangle is bh, where b is the length of the base and h is the height. This method and others are discussed in full in Area of Triangles. Trigonometry, however, provides additional ways to find the area of a triangle using the trigonometric functions. There are three basic situations in which the area of a triangle can be calculated using trigonometric techniques.
Two Angles and a Side are Given
If two angles and a side are known, the third angle can be calculated. Once it has been calculated, the following formula can be used to calculate the area of the triangle:
Two Sides and the Angle Opposite are Given
If two sides and the angle opposite one of them is given, the Law of Sines can be used to calculate the value of a second angle. The third can be calculated using subtraction, and at that point, the formula above is usable. Remember that sometimes the triangle found using these techniques is ambiguous, so you may have to find the areas of both possibilities.
Two Sides and Their Included Angle are Given
Given two sides and their included angle, the following formula can be used to find area:
|Area = ab sin(C)|
With these formulas, in addition to the traditional geometric formulas and Heron's Formula, you have enough techniques to calculate the area of almost any triangle, provided that something is known about it.