Question

How to design an algorithm to find minimum no.of sides of a polygon which lies between two concentric circles?

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Answer 1

Think about the simplest case first: the inner circle is microscopically small. The minimum number of sides is 3, as long as the inner circle has a non-zero radius.

When does the polygon start needing 4 sides? Draw an equilateral triangle inscribed in the circle. The polygon starts needing 4 sides when the inner circle’s radius reaches the center point of the sides of the triangle.

If you inscribe a regular polygon of N sides into the outer circle, you can compute the distance from the midpoint of each side to the center of the circle using the cosine rule:

distance_to_midpoint = cos ( 360 / (N * 2) ) * radius_of_outer_circle

(Explanation: if you make a isosceles triangle using the center point of the circle to the side in question, the radii have an angle of 360 / N. Divide the triangle in half at the side’s midpoint to form a right-angle triangle with hypotenuse equal to the radius of the outer circle, then use cosine rule)

Now distance_to_midpoint needs to be greater than or equal to the radius of the inner circle, so solve for N:

radius_of_inner_circle = cos(360 / (N * 2)) * radius_of_outer_circle
cos(360 / (N*2)) = radius_of_inner_circle / radius_of_outer_circle
360 / (N*2) = acos(radius_i / radius_o)
N = 180 / (acos(radius_i / radius_o))

(I haven’t double checked this math, and it’s really late).