Area of a Triangle Using Sine

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Let the triangle have side lengths , and . Then the area is . A proof is outlined in the Details.

Contributed by: Enrique Zeleny (July 2018)
Open content licensed under CC BY-NC-SA



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 )

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