# Would a $\infty$ sided regular polygon describe a circle?

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?

• A circle is defined by its radius, so a fairer comparison would be between the apothem of a polygon and the radius of a circle, and not one using interior angles, since a circle has none. – Frpzzd Apr 30 '17 at 21:31
• The first thing one should ask in such discussions is: "What is a side?". – Ennar Apr 30 '17 at 21:38

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.
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.