11 comments on “How Many Sides Does A Circle Have?

  1. Pingback: This Week at the Global Math Department / Global Math Department

  2. Your comments on the possibility of a circle having sides and the theory our minds need to suggest such a thing are really helpful. I am new to Mathematics at the age of 71 but all these great t.v. documentaries make me think….Juliet

  3. The square only has more than four sides if you allow duplicate points, which seems wrong. There is one point at each corner. Add an infinite number of identical points to the same corner, it is still effectively one point. A circle does have an infinite number of points because they are all in different locations. Hence a circle has an infinite number of sides.

  4. In support of the infinite sides:

    The length of sides of an infinite sided regular polygon (apeirogon) is not zero. It is infinitesimally small. Such a polygon has infinite sides and infinite corners. The sides and corners are not the same in such a polygon. Speaking in terms of limits, the length of sides approaches zero & the number of sides approaches infinity but they do not get there.

    About the octagon-to-square limit argument, since the sides are being constructed in an infinite way, we need to consider the defining the number of “sides between two points” to be based on the number of extreme points of the construction that lie outside of a bounding rectangle whose width is shrinking to zero. Under such a definition, the sides in the construction taken as a limit will tend from 8 to 4 and will converge. It can be any other similar limit based definition of a “side” which agrees that octagon has 8 sides & square has 4 sides – and per any such definition, a circle will have infinite sides.

    The very concept of a circle implicitly assumes the concepts of infinities, infinitesimals & limits. If circles do exist, there must be a way to construct them. Any such construction will lead us to the infinite sides argument.

  5. but what if you think about it with the angle differentiation like each side is an infinitly small angle differance instead of the size of the side

  6. This is so helpful! I am a philosopher, and my newly turned 5-year-old is a passionate defender of “circles have one side,” while I used to tend toward “circles have no sides,” but now I will forever approach the question with “what do you mean by ‘side'”? I’m looking into your “The Math Handbook” for my kid–he seems to be a math whiz–hasn’t even started school yet and regularly stumps me with philosophical math questions like this and “is zero even or odd? What about infinity?” and he argues vehemently that there must be a biggest number. (The book will be for both of us :))

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