given: big circle with radius 1, small circle with given radius r/1 {0<r<1}
find: fewest-sided regular polygon inscribed in bigger circle that does not cut the small circle

brief commentary

• hey guys
• i was told this problem would require calculus (it did not)
• just use radians lol
• AB (defined from the perpendicular bisection of one of the n-gon's sides) is always the smallest line from the center to the inscribed n-gon's perimeter (very easy to prove), so the maximum radius given the n-gon will always equal the length of AB (left column, bottom line) so i ignored the other circle
• putting that bottom-left formula for teh maximum into terms of AB will give us a minimum number sides (trust me)
• ceil on the last step bc i dont care if youve defined a way for polygons to have a non-integer number of sides go away
• desmos with graph of solution + function for max smaller radius for n sides