Problem 6 · The cushion in the circle
Sliding arcs preserves area: the cushion is exactly $2r^2$.
Integer answer, at most 4 digitsA circle of radius $10$ cm has been divided into four equal arcs. Then two arcs equal to the dashed ones have been drawn to produce the figure on the right. What is the area of the shaded region?
Reasoned solution
Key idea: between a chord and a quarter-circle arc there is always the same circular segment: replacing the top arc by its flipped twin trims exactly two segments per side.
Call $A$, $B$ the endpoints of the top arc. The circular segment between chord $AB$ and the ($90°$) arc is
At the top, the cushion loses (relative to the circle) the segment of the original arc and that of the new flipped arc (which is identical): $2s$. The same happens at the bottom. Hence:
Remarkably, $\pi$ cancels out: the cushion's area is exactly twice the square of the radius.