"Samuel B." <stb### [at] hotmailcom> wrote:

> It looks like a good solution, but I must admit I can't quite comprehend a lot
> of it. (I will keep trying, though.)

Apparently it is "the" solution, as I basically came across this exact equation
once when I was trying to save myself the trouble of re-deriving it.   :D

You have the "height" h of the arc, or depth that the circle/sphere pokes into
the object when it butts up against the edges.   Picture resting an oversize
ball in the mouth of a glass.

You also have the radius of the desired circle. r

We draw a chord perpendicular to the radius at height h, and use one half of
that as the leg of a triangle.  (L/2)  Drawing a line from the edge/circle
intersection to the circle center, we get the hypotenuse, which is just another
radius of the circle.

The other leg of the triangle is the radius minus the height of the arc.

Now you just do the Pythagorean theorem, and expand terms, simplify, and
rearrange to get r.

The diagram is messier because I used (L/2) instead of defining l = L/2, which
just makes the equation look a lot nicer.

Hope that helps.

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