This paper investigates the finite-time coordinated circumnavigation problem for multiple agents. The agents are required to surround a moving target with prescribed circular velocity and formation under velocity constraints. All agents can communicate to their neighbors by a digraph, while the topology of the sensor graph between agents and target may dynamically switch. We first design a distributed finite-time observer to estimate the state of the target. Then, a distributed observer-based control strategy is proposed, so that all agents can converge to their desired trajectories in finite time. Finally, numerical simulations illustrate the effectiveness of the proposed methods.