Good day, my family had a dinner discussion about polygons and how many sides a polygon has in relation to the angle measurement you'll get when you measure an "arc" encompassing a "side" of the polygon. Of course this is assuming that all sides are equal.

In this case, we'll get 120 degrees for a triangle, 90 degrees for a square, and the degree measurement decreases as the number of sides increase.

Now, my question is, with this above in mind, up to how many sides can a polygon have? We can have a 360-sided polygon with each side having an arc of 1 degree, but you can go smaller than 1 degree and argue that you can go with 0.000001 degree and have the corresponding number of sides. However, since there are an infinite number of fractions between two numbers, we can go infinitely many times and get n amount of sides based on a very small angle m.

If we go as such, up to how many sides can we get before the shape we have can be considered as a circle?


closed as off-topic by Xander Henderson, Namaste, max_zorn, Lord Shark the Unknown, Michael Rozenberg Oct 27 '18 at 6:34

This question appears to be off-topic. The users who voted to close gave this specific reason:

  • "This question is not about mathematics, within the scope defined in the help center." – Xander Henderson, Namaste, max_zorn, Michael Rozenberg
If this question can be reworded to fit the rules in the help center, please edit the question.

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    $\begingroup$ what is "considered as a circle"? other formulation: "what is a circle?" Let $P$ be any regular polygon inscribed in a circle, then take the median of each edge (orthogonal line passing by the middle), then they cross the circle in some points equidistant on the circle. Connect these points to get a polygon $P'$ which has two times more vertex (but is still not a circle). This shows that you can build regular polygons with $n$ edges for any $n\in\Bbb N$ (and it is still not a circle). $\endgroup$ – Surb Mar 12 '15 at 13:29
  • $\begingroup$ Considered by whom? $\endgroup$ – MJD Mar 12 '15 at 13:35
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    $\begingroup$ I wish my family had discussions about polygons $\endgroup$ – Hrodelbert Mar 12 '15 at 13:38
  • $\begingroup$ A side note: a regular $n$-polygon inscribed inside the unit circle has perimeter $2n\sin(\pi/n)$ and area $(n/2)\sin(2\pi/n)$ both of which approach the corresponding values for the circle as $n\to\infty$. $\endgroup$ – Kim Jong Un Mar 12 '15 at 17:42
  • $\begingroup$ Psychological question you've asked, regarding "perception". That's not a topic for Math.SE. $\endgroup$ – Namaste Oct 26 '18 at 19:57

A polygon is never a circle, no matter how many sides it has.

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    $\begingroup$ There may be a number of sides such that our perception cannot tell the difference: But that would be a psychological question not a mathematical question. $\endgroup$ – Circulwyrd Mar 12 '15 at 13:41

You and your family are effectively exploring the notion of limits from calculus in a geometric context. Think of this problem in terms of real numbers for a second. If we start at 1 and add $\frac{1}{2}$, and then add $\frac{1}{4}$, and then add $\frac{1}{8}$, etc., how many times would we need to add these terms before the sum was considered to be the number 2? The answer is that for any finite amount of these terms that we add together, our sum will always be less than two. Even so, we can observe that no matter what small number we choose, there will be one of these sums will be less than that small number away from 2.

Now apply this idea to regular polygons and circles.


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