 # 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

## Snapshots   ## 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 )

## Permanent Citation

Enrique Zeleny

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