The law of cosines, apparently not well known, was used in a previous post...

Stepping off a polygon

to good effect to solve a not uncommon shop problem.

Here we use a somewhat more obscure theorem to unravel another shop problem.

Suppose you have a circular segment like the figure APBXA in the illustration below. It could be a gear fragment, a key segment you want to reproduce, or even a flower bed you want to extend. How do you go about determining the radius of the circle from which the segment derives?

The tool we'll use is the intersecting chords theorem. It says that, if two chords intersect inside a circle, their individual parts are mathematically related to each other. A full description, including a proof of the theorem, is here...

https://en.wikipedia.org/wiki/Inters...chords_theorem

From the intersecting chords theorem, we have:

XP x XQ = XA x XB

Define:

c = AB ; c = chord length of segment

h = XP ; h = "height" of segment

which are things we can measure from the segment, then, substituting into the intersecting chords theorem, we have:

(c/2)^2 = h x XQ

which can be solved for XQ

XQ = c^2/(4h)

The diameter, PQ = PX + XQ and

PX + XQ = h + c^2/(4h) = (4h^2 + c^2)/(4h)

and the radius of the circle, CQ, is half this diameter

r = CQ = (4h^2 + c^2)/(8h)

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