# General Formula for Area of a Triangle

With the help of sine, we can generalize the formula for the area of a triangle to work for any triangle. The formula looks like this:

Formula

### GeneralFormulaforAreaofaTriangle

$\begin{array}{lll}\hfill T& =\frac{1}{2}\cdot b\cdot c\cdot \mathrm{sin}\angle A\phantom{\rule{2em}{0ex}}& \hfill \text{(1)}\phantom{\rule{0.33em}{0ex}}\\ \hfill \angle A& ={\mathrm{sin}}^{-1}\phantom{\rule{-0.17em}{0ex}}\left(\frac{2T}{b\cdot c}\right),\phantom{\rule{2em}{0ex}}& \hfill \text{(2)}\phantom{\rule{0.33em}{0ex}}\end{array}$ where $b$ and $c$ are the sides meeting in angle $\angle A$, and where $T$ is the area of the triangle $△ABC$.

Rule

### Uses

You can use this generalized formula for the area of a triangle to

• Find the area of a triangle where you know two sides and the angle between them. Formula (1).

• Find an angle in a triangle where you know the area and the two sides forming the angle. Formula (2).

Example 1

Find the area of a triangle with sides $b=30$ and $c=15$ that meet in an angle $\angle A=\text{}135\text{}\text{°}$

Insert the numbers into a calculator and compute:

 $T=\frac{1}{2}\cdot 30\cdot 15\cdot \mathrm{sin}135\text{°}\approx 159.9.$

Example 2

Find the angle $\angle A$ in a triangle $△ABC$, where $\angle A$ is formed by the two sides $b=10$ and $c=5$, and the area of the triangle is $T=\text{}12.5\text{}$

You can insert this information into Formula (2) in `CAS`:

 $A={\mathrm{sin}}^{-1}\phantom{\rule{-0.17em}{0ex}}\left(\frac{2\cdot 12.5}{10\cdot 5}\right)=30\text{°}.$