Picture describing the puzzle What is the area of the region poly1 formed by the arcs 'cdke'. The square is of sides 10 units long. The region poly1 is formed by four overlapping quadrants.


closed as off-topic by rhsquared, Excited Raichu, Oray, JMP, athin Dec 8 '18 at 17:29

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  • 1
    $\begingroup$ Ignoring the fact that this is puzzling.se This question can't be solved purely mathematically. I mean given a square of length 10 I can draw many different arcs cdke such that the inner area is anything from almost everything to almost nothing. $\endgroup$ – Yanko Dec 8 '18 at 14:18
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    $\begingroup$ I assume the arcs are quarter circles centered on the respective corners of the square? $\endgroup$ – SteveV Dec 8 '18 at 14:33
  • $\begingroup$ @Yanko can't it be solved with the inclusion–exclusion principle? $\endgroup$ – Weather Vane Dec 8 '18 at 15:13
  • $\begingroup$ @WeatherVane I guess this is the idea, but right now you need some more assumptions about the question. Perhaps with SteveV's extra assumption it is possible to solve this question. $\endgroup$ – Yanko Dec 8 '18 at 15:16
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    $\begingroup$ ...oh I thought that was #obvious# especially as they are called "quadrants" and there is presumably one solution. $\endgroup$ – Weather Vane Dec 8 '18 at 15:18

Partial Answer:

Notation: notation

Step 1:

Adding the four quadrants would render
2(I+III+VII+IX) + 3*(II+IV+VI+VIII) + 4*V = pi*10*10

Step 2:

Minus 2 10*10 square from both sides:
1*(II+IV+VI+VIII) + 2*V = 100(pi-2)

Step 3:

II+V+VIII = pi*10*10/2 - 10*10 = 100(pi/2-1)
(2 quadrants minus square)


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