Other correct solutions were sent in by Ray, Kremer, David Aberle, Andrew Hess.  Further solutions were submitted by Ralf Gesellensetter, Dan Jane, Pat Collins, Philippe Fondanaiche, Burkart Venzke, Ariel Flat.

Carl Westine solved the problem for any piece of pie having angle measure 0 < q £ p/2.  For the specific case mentioned in the problem, the piece of pie will have angle measure p/4.  Consider the diagram on the left.  Draw a line dividing the pie into two pieces having angle measure p/8.  This line will pass through the center, O, of the circular plate of unknown radius R.  Drop a perpendicular from this center as shown.  This will bisect the chord AB from which you get the equation cos p/8 = .5/R which yields R = .5411...  The general formula for angles in the range above is R = 1/(2 cos(q/2)).  Interestingly, this formula is not correct for angles larger than p/2.  It's easy to see that if q is larger than p, then R = 1.  However, in the region p/2 < q < p the required radius is not given by the formula above.  Can you figure out, as Andrew Hess did, what goes wrong?

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Page last updated 12 October 2000.