Area of a Triangle Using Sine
Let the triangle have side lengths , and . Then the area is . A proof is outlined in the Details.
Here is a derivation of the formula. Draw a perpendicular from the point to the side at . The triangle is now divided into two right triangles and . Let . Let the lengths of the two segments of be and .
Then, by trigonometry, , , . (*)
The area of the triangle is
(the base is and the height is )
(substituting from (*))
(factoring out )
(using the expansion of the sine of a sum in reverse)
. (adding the two angles at )