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## Question

Suppose i have a circle. I cut off its arcs such that it became the biggest possible square i could make from that circle.  What's the ratio of the edge of the circle to the middle of the edge of the square (assume minimum length) to the radius of the circle.

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Suppose i have a circle. I cut off its arcs such that it became the biggest possible square i could make from that circle.   This is clear

Several clarifications are required:    What's the ratio of the edge of the circle .....(1)  edge of the circle  a) the circumference    or b)  one of the arcs

clarification required:  to the middle of the edge of the square   ?                         (assume minimum length) to the radius of the circle.

. Is the question : What is the ratio of the arc length   to the perpendicular length from the center of one side of the square to the arc ???

Suppose i have a circle. I cut off its arcs such that it became the biggest possible square i could make from that circle.  What's the ratio of the edge of the circle to the middle of the edge of the square (assume minimum length) to the radius of the circle.

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If the question is : What is the ratio of the arc length   to the perpendicular length from the center of one side of the square to the arc ???

Then assuming diameter of 1" ; the diagonal of the largest square would also be  1 ; this divides the square into  2 isosceles triangles ,whose angles are 45,45,90   In a 45 degree triangle the sides are = Hsin angle  or 1 sin 45 = 0.707;  therefore i/2 of the lt of a side is 0.353 .    The radius is 0.5.   0.5 - 0.353 = the rise  0.147  the distance from arc to square

Now the circumference  is pie D or 3.14 D,  3.14 (1) = 3.14;  acr length 3.14 divided by 4 =0.7853  ratio of arc length to rise  ....0.7853/0.147 =  5.342 :1

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Spoiler

Assume the radius R = 1. The diagonal of the square will be 2R = 2. The sides of the square will be sqrt(2). The length from the center to the midpoint of the square's side is sqrt(2)/2. The ratio is then (1-sqrt(2)/2) / 1, or just 1 - sqrt(2)/2, or about 0.3.

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If the question is : What is the ratio of the arc length   to the perpendicular length from the center of one side of the square to the arc ???

Then assuming diameter of 1" ; the diagonal of the largest square would also be  1 ; this divides the square into  2 isosceles triangles ,whose angles are 45,45,90   In a 45 degree triangle the sides are = Hsin angle  or 1 sin 45 = 0.707;  therefore i/2 of the lt of a side is 0.353 .    The radius is 0.5.   0.5 - 0.353 = the rise  0.147  the distance from arc to square

Now the circumference  is pie D or 3.14 D,  3.14 (1) = 3.14;  acr length 3.14 divided by 4 =0.7853  ratio of arc length to rise  ....0.7853/0.147 =  5.342 :1

Previously I assumed the wrong question ,However assuming a diameter of 1"   R = 0.5"  .  I calculated the rise ( r ),which is the distance from the one side of the square at the mid point of that side to the arc or circumference . That value  r =0.147  therefore the ratio of r/R  = 0.147/0.5       0.147 /0.5  = 1/X;             0.147X = 1x 0.5;    X = 1 /0.147;

Ratio r / R    is    1 /6.802

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From my reading of the problem, the question can be re-phrased:
What is the ratio of the sagitta to the radius of a circle where the apothem is equal to the square-root of half the radius squared?

Spoiler

The chosen answer is valid for radius = 1, yet my calculations do not show that to be true for other radius lengths.

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