Given a square, construct a region in the center that is bounded by the four quarter-circle arcs using the vertices as centers and sides of the square as radius.
Find the area of that center region -- the yellow region in the figure at the right -- for a square with side length s.
Hint: There are three different shapes in the square -- yellow, pink, and blue. If we add the areas of each of the four Quarter-Circles we get an area that is equivalent to 4 yellow, 12 pink and 8 blue. Now 3 yellow + 12 Pink + 12 blue would be 3 squares
So the area of 1 yellow is the total of 4 quarter-circles - 3 squares + 4 blues. Try to find an expression for one of the blue sections in terms of s.