We consider the problem of navigating a mobile robot that is located at any arbitrary point within a bounded environment, to a point on the environment's outer boundary and then, using the robot to explore the perimeter of the boundary. The environment can have obstacles in it and the location and size of these obstacles are not provided a priori to the robot. We present an online path planning algorithm to solve this problem that requires very simple behaviors and computation on the robot. We analytically prove that by using our algorithm, the robot is guaranteed to reach and explore the outer boundary of the environment within a finite time.