Formula for the area of the interior of a closed contour in the complex plane

Question

Let $\gamma \colon [a,b] \to \mathbb{C}$ be closed contour in the complex plane. What is the formula for the area of the interior $I(\gamma)$ of $\gamma$? My assumption is that it involves some kind of contour integral. If possible, please kindly show me a derivation and direct me to a resource where I can research this further. Many thanks.

Answer 1

Note that applying Green formula for $\bar z dz$ we get that the area inside $\gamma$ (assuming the curve is simple and piecewise smooth say) is $\frac{1}{2i}\int_{\gamma}{\bar z }dz$