Triangle Area with Half ab sin C

Half base times height always works for area, but you need the height. When nobody hands you a height, build one. Take a triangle with sides a and b and the angle C between them. Use side a as the base. Drop a perpendicular from the far corner to the base, making a right triangle whose hypotenuse is b and whose height is h. In that right triangle, h equals b sin C. Substitute into the base-height formula and the area is one half times a times b times sin C. The rule is that a and b must touch angle C: grab a far side and the answer breaks.

When angle C is obtuse, the perpendicular meets the extension of the base instead of the base itself. The auxiliary triangle still has hypotenuse b and an acute angle of 180 minus C, and sin of 180 minus C equals sin C, so the height is unchanged. At exactly 90 degrees the perpendicular is side b itself, which is the right-angle case of the same formula.

The formula also runs backwards. When you know the area and two sides, solve for the sine of the angle between them: sin C equals 2 times area over a times b, then take the inverse sine. Sine only reaches 1, so the area can never exceed half of a times b, and that maximum happens only at 90 degrees. When sin C is exactly 1 both candidates coincide at 90 degrees, counted once. For any smaller sine, both C and 180 minus C are possible, check which one fits your triangle.