The typical way students remember the area of a triangle is

However, there are other formulas for the area of a triangle which can be helpful if the height is not immediately known.

**Case 1: SAS.** Suppose that two sides and the angle between the sides — say, and and the measure of angle — are known.

If is an altitude for , then is a right triangle. Therefore,

, or .

Therefore,

.

Using the same picture, one can also show that

Also, with a different but similar picture, one can show that

An important consequence of the SAS area formula is the *Law of Sines*. Since all three formulas must give the same area , we have

Multiplying by produces the Law of Sines:

Case 2: ASA. Now suppose that we are given the measures of two angles and the length of the side in between them — say, angles and and side . Naturally, we can also get the measure of angle since the sum of the measures of the three angles must be .

From the SAS formula and the Law of Sines, we have

Combining these, we find

By similar reasoning, we can also find that

and