Area of Infinite Pointed Star
Suppose we have 2 concentric circles with radii and for some integer . Place equally spaced points on the circumference of the larger circle. For each such point, draw two tangent line segments to the smaller circle and shade in the area enclosed by the new line segments and the smaller circle.
As tends to infinity, what portion of the larger circle do we shade with the above construction?
CLARIFICATION: once a region is shaded, shading it again has no effect.
note: sorry there's no picture... message me if you know of a good site to draw geometry and export images!
0 but <1
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1 solution
As n tends to infinity, the length of the points on the star tend toward π + 1 π , and roughly (using "roughly" liberally) as much area is covered by star points as by the space in between as they are interlocking triangles. Thus there is asymptotically nonzero area shaded as well as nonzero area non-shaded, meaning the portion shaded is strictly between 0 and 1.
The real answer is 0.433