# Formulas for the Area of a Triangle

1. $A = \frac{1}{2}b \cdot h$, where $b$ is the base and $h$ is the altitude of the triangle.

2. The area of an equilateral triangle $A = \frac{{\sqrt 3 }}{4}{a^2}$, where $a$ is the length of each side of the triangle.

3. The area of a triangle when two adjacent sides and the included angle is given by $\Delta = \frac{1}{2}a \cdot b\sin \theta$

4. The area of a triangle when the length of all sides are given $\Delta = \sqrt {s\left( {s – a} \right)\left( {s – b} \right)\left( {s – c} \right)}$ where $s = \frac{{a + b + c}}{2}$

5. The area of a triangle with vertices $A\left( {{x_1},{y_1}} \right),\,B\left( {{x_2},{y_2}} \right),\,C\left( {{x_3},{y_3}} \right)$ is given by the formulas $A = \frac{1}{2}\left| {\begin{array}{*{20}{c}} {{x_1}}&{{y_1}}&1 \\ {{x_2}}&{{y_2}}&1 \\ {{x_3}}&{{y_3}}&1 \end{array}} \right|$