Would a $\infty$ sided regular polygon describe a circle?
Emily Wong 
Help us settle this argument. Consider a circle. What would best describe a circle?
A zero sided regular polygon? The other person thinks this because it cannot be infinite, because the shape is enclosed. There is no end, and there has to be a limit on size.
I disagree with this because the size can be made infinitely small so that it can have infinite sides. Also, zero sides= nothing
An infinite sided polygon I think this because a tangent has a angle of 180 where it meets an edge, and for a $n$ sided polygon, the interior angle converges to 180 when $n$ goes towards $\infty$
The other person disagrees with this because apparently, size cannot be unlimited.
Could you say who is right?
$\endgroup$ 24 Answers
$\begingroup$A circle is not a polygon with infinite sides, and it's not a polygon with zero sides. It's not a polygon at all. This boils down to an argument about terminology - what does it mean for something to have an infinite number of sides? This would have to be precisely defined before the question makes sense. However, what's clear is that as the number of sides of a regular polygon tends to infinity, the shape approaches a circle (although there are a number of ways to interpret this statement in precise mathematical terms.)
$\endgroup$ 2 $\begingroup$If you consider the sequence of regular polygons with $n$-sides and inscribe them in a circle. You can see quite clearly that, under the limit as $n\rightarrow\infty$, the polygons are approaching the circle.
$\endgroup$ 1 $\begingroup$One way to settle this would be to use limits. For example, the formula for the area of a regular polygon of $n$ sides given its apothem $a$ is $$a^2n\tan\frac{180}{n}$$ The limit of this as $n$ approaches infinity is given by $\pi a^2$, so I would say that area-wise, the analogy seems to make sense. You could use a different limit to draw a comparison between the circumference formula.
$\endgroup$ $\begingroup$In the infinite side case, each side is a point, right? Since in a regular polygon all the vertices are equidistant from a central point, then we have an infinite collection of points which are vertices, thus an infinite collection of points equidistant from a central point, which is exactly a circle I would think.
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