Try this problem from ISI B.Stat, B.Math Subjective Entrance Exam, Problem 4 based on Region close to the center.

Problem:

Let S be the square formed by the four vertices (1, 1), (1, -1), (-1, 1), and (-1, -1). Let the region R be the set of points inside S which are closer to the center than to any of the four sides. Find the area to the region R.

Nice analysis sir