I think it would be really interesting if you could approximate the area (and perhaps circumference) of the unit circle using a n-sided regular polygon inscribed in that circle. If you can find a general function, say A(n), that represents the area of the polygon with n sides, then the limit as n approaches infinity should be pi!

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Perhaps in addition to this you could look at why using an n sided polynomial to approximate a circle would work for area but not for perimeter.