Number of regions inside a circle
Geometry
Level
3
For a regular 16-gon inscribed in a circle, how many separate regions can be generated by the diagonals?
Examples for a diameter, an equilateral triangle, and a square are shown below:
1696
1712
32768
65536
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1 solution
The number of regions inside a regular 1 6 -gon is 1 6 9 6 . This number is given by OEIS , and a formula for the regular n -gon which proves the result for n = 1 6 is given on page 3 of this article . Adding 1 6 for the regions between the polygon and the circle, we obtain 1 7 1 2 regions inside the circle.