Deriving the Area of a Triangle using Trigonometry

The area of the triangle is half the product of its base and altitude. In the figure below for instance, the area  of the two triangles with base b and altitude h is given by the formula Area = \frac{1}{2}bh.

area of a triangle

 

In this post, we derive the area of a triangle using trigonometry In particular, we want to compute for the area of triangle ABC given angle C and side AC.

triangle-2

 

Given C and side AC, we can compute for the altitude since

\sin C = \displaystyle \frac{AM}{AC}.

Simplifying, we have

AM = AC \sin C = b \sin C.

 Now, since area is the product of its base and its height,

Area = \frac{1}{2} BC \times AM.

Area = \frac{1}{2} (ab) \sin C.

From this derivation, it also follows that

Area = \frac{1}{2} (ac) \sin B

and

Area = \frac{1}{2} (bc) \sin A.

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