When You Know 2 Sides and 1 Angle
\begin{equation*}
A=\frac{1}{2} b c \sin \alpha
\end{equation*}
When You Know 1 Side and 2 Angles
\begin{equation*}
\frac{a^{2} \sin \beta \sin \gamma}{2 \sin \alpha}
\end{equation*}
Proof:
Utilize the law of sines and plug into the previous formula.
Hero’s Formula
\begin{equation*}
A=\sqrt{s(s-a)(s-b)(s-c)}
\end{equation*}
Proof:
Start with:
\begin{equation*}
A=\frac{1}{2} b c \sin \alpha
\end{equation*}
Then:
\begin{equation*}
\begin{array}{l}=\frac{1}{2} b c \cdot 2 \sin \frac{\alpha}{2} \cos \frac{\alpha}{2} \quad\left(\sin \alpha = 2 \sin \frac{\alpha}{2} \cos \frac{\alpha}{2}\right) \\ =b c \sqrt{\frac{(s-b)(s-c)}{b c}} \sqrt{\frac{s(s-a)}{b c}} \text { (by half angle formulas) } \\ =b c \frac{\sqrt{s(s-a)(s-b)(s-c)}}{b c}\end{array}
\end{equation*}