Infinite Webbed Heart

Calculus Level 5

The three figures below are made of regular hexagons, regular heptagons and regular octagons respectively.

In general, if this pattern continues, we get a heart-shaped figure by drawing n 1 n-1 regular n n -gons among which the smallest one can be inscribed in a circle with radius of π n . \frac{\sqrt\pi}n. What is the area of the total figure as n n approaches infinity?


Inspiration


The answer is 1.5.

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