  We prove a sharp stability result for the Brunn-Minkowski inequality for
$A,B\subset\mathbb{R}^2$. Assuming that the Brunn-Minkowski deficit
$\delta=|A+B|^{\frac{1}{2}}/(|A|^\frac12+|B|^\frac12)-1$ is sufficiently small
in terms of $t=|A|^{\frac{1}{2}}/(|A|^{\frac{1}{2}}+|B|^{\frac{1}{2}})$, there
exist homothetic convex sets $K_A \supset A$ and $K_B\supset B$ such that
$\frac{|K_A\setminus A|}{|A|}+\frac{|K_B\setminus B|}{|B|} \le C
t^{-\frac{1}{2}}\delta^{\frac{1}{2}}$. The key ingredient is to show for every
$\epsilon>0$, if $\delta$ is sufficiently small then $|co(A+B)\setminus
(A+B)|\le (1+\epsilon)(|co(A)\setminus A|+|co(B)\setminus B|)$.
