In this paper, we study the solutions to the optimal exit time control problem. Such a problem tries to find the state feedback control law with a fixed cost that can keep the state of a randomly perturbed system inside a subset of the state space, called the safe set, for as long as possible on average. By formulating the problem as an optimization problem with PDE constraints and using symmetrization techniques, we show that, when the safe set is a ball, the optimal feedback control (if it exists) must be radially symmetric. Furthermore, we show that, among all safe sets with a fixed volume, the ball is the best in that it yields the most efficient optimal exit time control. The proofs make essential use of the general isoperimetric inequality.