In
the illustration above, A is the centerpoint of the red circle. The red circle
is reflected across the yellow lines in four directions to create the ring of
overlapping identical circles around the red one. Each respective reflection of the original
red circle is marked by their center points-A’, A’1, A’2, and A’3. I added a
quadrilateral (the rotated blue square) drawn between the four center points of
each outer circle, and the space marked by Inner consists of all the area not
overlapped by the reflection circles. Given that the blue quadrilateral square
has a perimeter of 16, is it possible to identify the Inner area of the object?
The area of the inner star-like thing is
ReplyDelete8-4(pi-2)
The hypotenuse of the blu-yellow triangle with the red arc going through it is sqrt8, meaning that the area of the yellow square is 8.
If the area of the black circles is 4pi then a quarter of that is pi (the blue square cuts into the circles and forms a quarter). If the area of the quarter is pi and the area of the triangle is two, then to find the semi-circle thing subtract and get pi-2. There are four of these semi-circle things inside the yellow square meaning that it is 8{area}-4*(pi-2)
All work on this answer was done by Jack Fergus™