  A pair $(\alpha, \beta)$ of simple closed geodesics on a closed and oriented
hyperbolic surface $M_g$ of genus $g$ is called a filling pair if the
complementary components of $\alpha\cup\beta$ in $M_g$ are simply connected.
The length of a filling pair is defined to be the sum of their individual
lengths. In \cite{Aou}, Aougab-Huang conjectured that the length of any filling
pair on $M$ is at least $\frac{m_{g}}{2}$, where $m_{g}$ is the perimeter of
the regular right-angled hyperbolic $\left(8g-4\right)$-gon.
  In this paper, we prove a generalized isoperimetric inequality for
disconnected regions and we prove the Aougab-Huang conjecture as a corollary.
