State the isoperimetric inequality in the plane.

Let $S \subset \mathbb{R}^{3}$ be a surface. Let $p \in S$ and let $S_{r}(p)$ be a geodesic circle of centre $p$ and radius $r$ ( $r$ small). Let $L$ be the length of $S_{r}(p)$ and $A$ be the area of the region bounded by $S_{r}(p)$. Prove that

$$4 \pi A-L^{2}=\pi^{2} r^{4} K(p)+\varepsilon(r),$$

where $K(p)$ is the Gaussian curvature of $S$ at $p$ and

$$\lim _{r \rightarrow 0} \frac{\varepsilon(r)}{r^{4}}=0 .$$

When $K(p)>0$ and $r$ is small, compare this briefly with the isoperimetric inequality in the plane.