The area of a triangle is found by using the formula, where the base and an altitude of the triangle is required ; however trignometry offers us a formula which does not require a base or height. We need any two sides and an inscribed angle.

The deservation:

To derive our formula, we begin with the general formula for the area of a triangle:

The base of the triangle ABC in Figure 1 is c and the height is h. So the formula for A becomes, in this case,

Figure 1

Suppose that, for triangle ABC, we are given the lengths of sides b and c and the measure of angle A. then we can write sin A as

or, by solving for h,

Substituting this expression for h into the formula

we have

Applying the same kind of reasoning to the heights drawn from A and C, we also have

Example 1:

Find the area of triangle ABC if a = 12 cm and b = 15 cm and A = 60^o .

Example 2:

Find the area of triangle ABC if a = 17 cm and b = 18 cm and A = 38^o .