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With this given. Determining the radius or square seems like it should be possible, but it just seems we do not have enough values to work with. We can create equations that equal each other, but they just yield infinite many solutions. What we must find is an equation that describes the value of a or b in which a solution of a or b can be found. We need an equation that gives a value of a or b that doesn’t include it with a variable of the radius or other part of the square. Or we must find the value of the radius. This seems like it should be able to be done by simple algebra, but the calculation is more complex than thought on first inspection.

Let’s start with what is known about a square. (It will help to have a compass handy to look geometric constructions.) A square is separated into 2 equal triangles with a 45 deg. angle. This angles cosine and sine are equal. (Both equal the radius of the circle.) So if this 45 deg. angle is drawn from the bottom left to the upper right of the square it would mark the diameter of the circle and show where the square’s upper right side ends.

Also in a square if a 30 degree angle is drawing from the center to the upper right, its aligned length is equal to the radius. Also it’s sine is equal to ½ , so the y distance from the center equals ½ the radius.

Some equations are:
(cos 30 * r)^2 + (½ *r)^2 = (r)^2
and
cos 45 * (sqrt (r^2 + r^2)) = r

These equations are derived from the given. There are more variations of them, but by themselves they don’t lead to an answer without variables.

Now let’s look at this angle relationship of the 30 deg. and 45 deg. angle and see how it can be applied to solve the radius (a + 1.25) or (½ b + 2). But my last attempt has to do with were the 30 deg. angle and 45 deg. angle intersect. Normally these angles will not intersect, but the 45 deg. angle is placed here ½ the radius below the 30 deg. angle. But unfortunately the answer is still in the form of the radius.

r - (r * cos 30) = 0.1340 * r

The length of 0.1340 is proportional to all circles. That length of 0.1340 is the length between 30 degrees and 45 degrees on a unit circle.

In Conclusion:

This one is a stinker. It seems like it is possible to solve on a hunch, but there are so many unknowns. That combined with the fact that most of the equations still rely on the radius is making it difficult to use the equations. Still I think it is only a problem to tinker with when you, the mathematician are bored. There may be something here, but after hours at looking at this problem, I haven’t found it. This is a problem you tinker with in your free time. But don’t forget we already have 2 solutions for finding the radius of the circle without the square. I just want to find a way to solve the square.

 

 

 

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