SOLUTION: A circle is inscribed inside a square of which the square has side length 1 unit and the area outside the circle is painted black. Inside the circle, a square is inscribed and the
Algebra ->
Surface-area
-> SOLUTION: A circle is inscribed inside a square of which the square has side length 1 unit and the area outside the circle is painted black. Inside the circle, a square is inscribed and the
Log On
Question 1095315: A circle is inscribed inside a square of which the square has side length 1 unit and the area outside the circle is painted black. Inside the circle, a square is inscribed and the area outside the square is painted white. Inside the smaller square another circle is inscribed And that continues infinitely many times. What is the ratio of black to white area? Thank you in advance! Answer by greenestamps(13200) (Show Source):
You can put this solution on YOUR website! Don't think that you need to find the infinite sum of all the black area and the infinite sum of all the white area. Since the same process is repeated infinitely, the ratio of black area to white area is just the ratio of black area to white area after the second square is inscribed in the first circle.
The black area at that time is the area of the first square, minus the area of the first circle. The first square has side length 1, so its area is 1. The first circle has radius 1/2, so its area is pi/4. So the black area at this point is
The first circle has area pi/4; the second square has area 1/2. So the white area at this point is
Then the ratio of black area to white area at this point -- and therefore the ratio of black area to white area in the "completed" infinite figure -- is
