Hi Ed. because O and P are centers of the circles, OP is the length of a radius of the circles. Notice that QP is ALSO a radius. So it also equals r. I hope that helps.
excuse me, but... since both circles have radius r, then if we name the top dot A and the bottom one B, then, OA=AP=PB=OB = r WHICH MEANS SQUARE! aka: the shaded region's area is r^2
You're correct that the 4 sides of the quadrilateral are of equal length. However, this does not necessarily mean that the shaded region is a square. Rather, this information guarantees that the shape is a rhombus (a quadrilateral with 4 equal sides). In order to be a square, the four angles would need to be equal to 90?. This is not the case: angle A = angle B = 60?while angle O = angle P = 150?.
1 Explanation