Area of a Triangle From Two Sides and the Included Angle

The old formula, area equals one half base times height, needs a height segment standing straight up from the base. Most triangles don't hand you that height directly. But if you know one side (call it the base) and the length of the other side plus the angle it makes with the base, you can build the height yourself: it's just the second side times the sine of that angle. Drop a perpendicular from the far vertex down to the base line, and that perpendicular, the second side, and the angle between them form a right triangle where sine does exactly its job: opposite over hypotenuse.

So Area = 1/2·base·height becomes Area = 1/2·a·b·sin(C), where a and b are the two sides and C is the angle trapped between them, the included angle. Swap in the numbers and it works for any triangle, not just right triangles, because sine handles the slant for you. This is why the angle has to be the one between the two given sides. Use the wrong angle and you're measuring a height that doesn't belong to that base.

This formula sits right next to the Law of Sines and Law of Cosines as one more tool for triangles with no right angle anywhere in them. It also quietly contains the right-triangle case: when C is 90 degrees, sin(C) = 1, and the formula collapses back to the familiar 1/2·a·b, since one side is already the height. Past 90 degrees, sine starts decreasing back toward 0, so a triangle can actually lose area as its included angle swings from 90 up toward 180, even with the two sides held fixed.