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What is the simplest way to calculate the area of the blue section of the circle's ring shown below using the following datas :

a) K is the center of the square ABCD,

b) Vertices A and C are on the outer circle line of the ring

c)$\frac{r}{KM} =\sqrt{5} - 1$

d) The area of the square ABCD is $80\text{ cm}^2$.

enter image description here

I think that this is useful formula $A=(r_1^2-r_2^2)\pi$ , where A is area of the ring.

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The data given is either insufficient or redundant. If it's meant to be complete without the diagram, it's insufficient, since the position of $M$ is entirely unspecified, so nothing can be inferred from c). On the other hand, if the diagram is meant to be taken into account, b) is redundant. –  joriki Sep 16 '11 at 6:59
@joriki,I have found relation between larger and smaller radius so don't know whether your assumption is correct. –  pedja Sep 16 '11 at 7:54
My comment did not depend on any knowledge or lack thereof of the ratio between the radii. I was merely noting that, since $M$ appears only once in the statements, c) cannot imply anything useful, since $M$ could be anything, and the ratio between $r$ and some arbitrary line segment is irrelevant. This is on the assumption that the statements were supposed to be self-contained and not rely on the diagram. If they rely on the diagram, there's no reason to include b), which has the same status as some other facts that one needs to derive from the diagram. –  joriki Sep 16 '11 at 8:06
My comment was not on whether or not this can be solved, but on how you presented the facts. It makes sense to either rely on the diagram and state what cannot be immediately derived from it in the text, or to make a self-contained statement of the facts and consider the diagram only as an illustration; it makes little sense to rely on the diagram but then state some facts like b) that are obvious from the diagram, but state other facts, e. g. that $M$ also lies on the outer circle. –  joriki Sep 16 '11 at 8:08
@joriki,I think that a fact that $M$ lies on the outer circle line implies from the diagram –  pedja Sep 16 '11 at 8:15

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