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(cord / sin(30)) + (cord / sin(30)) + x = Diameter

Take a measurement from the mirror of the square at 30 degree angle. Will it reach the upper and lower corners of the square?

We are concerned with the top radius or top of the line segment. The length at which intersects the horizontal segment is the length of which the 30 degree angle goes beyond the horizontal segment.

Add this length to the original horizontal segment and divide by the cos(30). Divide this length by 2 and you have the radius.

It should work with any angle though it might require sin and cos angle to equal the radius such as a 45 degree angle. The cos(30) is longer than r and a 45 degree angle, but so is the square that mirrors 30 degrees.

Take an angle from the radius at 90 degrees. From the segment, determine what x direction (cos) the angle crosses the segment. This should be the same length the bottom crosses the segment. (Hopefully) Take the cos of that length and determine what sin is needed to cross that length. Divide that length by the sin.

**The preceding did not hold true. However it is on the right path**


Even though without some new method the above wasn’t true the ideas that form an equation and hold true is the following:

With the mirrored angle starting at the upper corners of the square that surrounds the circle of unknown radius, the angles all intersect the corresponding mirrored angle in the middle of the x axis of the circle. So many angles to choose from (special values), they can be found were ½ the cosine below the given segment of the circle to have the same x value.


Why is solving this and finding an equation that describes it so valuable? This would mean we could find a cirlce only knowing 1 value. That value is the value of the segment.


(percentage of difference between 30 and 45 lengths) *sin(30) * cord = x + sin(45) * cord

[sin(45) * cord + x] / [sin(30) * cord] = percentage difference

3.14679488 - 2.225125 = 0.92167 bisector - r = difference of 45 length - 30 length

test for r = 2.225125
[sin(45) * cord + x] / [sin(30) * cord] = 1.92167

since sin(45) * cord + x is 1.92167 x larger than sin(30) * cord

x = (1.92167 * sin(30) * cord) - (sin(45) * cord)
x = 0.31716

[x / sin(30)] + [cord / sin(30)] = [x / sin(30)] + [1.25 / sin(30] = r

r = 0.6343 + 1.5858
= 2.22012

good way to estimate length of r given cord length

However needs to be tested.