Question number 16. A circle with a radius of 4 was divided into four adjacent parts using arcs of circles with radii of 2 as shown here in the picture. The perimeter of each part is then equal to which of the following. So the circle of radius 4 is the outside circle here in bold and the circles of radius 2 are here divided in halves. So let me draw in that smaller circle to illustrate what's going on here. In orange here if I were to pick the center then this is the full circle and we're only taking half of that and the region we're talking about, let me shade that in blue, for the perimeter is the following shape. So then the perimeter we are trying to compute is made up of this semicircle over here like that and this semicircle over here which would be exactly the perimeter of the orange circle that is in solid. So the orange circle has radius 2 so perimeter is equal to then 2 pi times the radius which is 2 so we obtain 4 pi and the other region we have to worry about is here this piece that encloses the blue region and that's one quarter of the perimeter of the large circle. So in red we have one-fourth of the large perimeter and that is equal to one-fourth and then 2 pi times the radius which we were told is 4. The 4 and the one-fourth cancel and so we end up with 2 pi again. And so together the orange pieces which combine to be the perimeter of the orange circle and a piece of the perimeter of the large circle in red we add those up to obtain the perimeter of the blue region and that is 4 pi plus 2 pi and that gives us together a perimeter of 6 pi and that is one of these parts so that would be 6 pi perimeter and the answer is C.

geometry problem

circle arcs

perimeter calculation

semicircle

quarter-circle