# 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 $Area = \frac{1}{2}bh$.

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 $AC$.

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

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

## 2 thoughts on “Deriving the Area of a Triangle using Trigonometry”

1. Also a good spot to show the relations of $\frac {sin(A)}{a} = \frac {sin(B)}{b} = \frac {sin(C)}{c}$.