All this complicated maths is doing my head in. Fascinating it might
be to some but it's not necessary.  There are two other methods:

1) Cut a disk to the diameter you want.  Cut a strip of paper longer
than you want.  Wrap the strip around the disc. At the point where the
strip meets the beginning make a cut. Flatten it out.  This is the
length needed.

2) Draw a long straight line on a piece of paper. Put a mark at one
end.  Cut a disc to the diameter you want.  Make a mark anywhere on
the circumference.  Match up the marks then roll the disc along the
line until the mark on the disc touches the line again.  Make another
mark on the line.  Again, this is the length needed.

Philip

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