User talk:Bwkollmar

Consider the rectangle bounding a single cycloid arch. From the definition of a cycloid, it has width $2πr$ and height $2r$, so its area is four times the area of the circle. Calculate the area within this rectangle that lies above the cycloid arch by bisecting the rectangle at the midpoint where the arch meets the rectangle, rotate one piece by 90° and overlay the other half of the rectangle with it. The new rectangle, of area twice that of the circle, consists of the "lens" region between two cycloids, whose area was calculated above to be the same as that of the circle, and the two regions that formed the region above the cycloid arch in the original rectangle. Thus, the area bounded by a rectangle above a single complete arch of the cycloid has area equal to the area of the circle, and so, the area bounded by the arch is three times the area of the circle.