This question has been bothering me of late. Does a circle have one side or an infinite number of sides? My teacher suggests that it is an infinite number of sides, her evidence being that the ratio between the perimeter of a polygon and the longest diagonal of a polygon tends towards pi as the number of sides of the polygon increases. The thing that has got me thinking is that if this is the case then as the number of sides increase, the length of the sides gets shorter and shorter until they have no length at all, i.e. they are points. But points are one dimensional, and then you can't have a line at all, because to the best of my knowledge you can't add two things in one dimension to get another dimension. Any thoughts? Or thoughts on how I should be thinking about this?

It depends on what you mean by "side". If you mean how many straight line segments are there, then a circle has none, as no segment of a circle is a straight line. An alternative definition of the number of sides (which makes more sense to me personally) would be the least number of differentiable (this means smooth, not angular) curves the shape can be divided into. Using this definition, a circle has 1 side. Here are a few more examples (badly drawn I'm afraid):



I don't think that using any particularly useful definition of side, a circle would have an infinite number of sides. However, it is true that the limit of an n sided regular polygon as n tends to infinity is a circle. In the limit each "side" is indeed a point, which is why I think that sides is no longer a useful word to describe them. Don't worry about adding things in one dimension to get another one. It is true that you cannot add two things in one dimension to get another dimension. However, this argument only works for what are called "countable" unions. For very large numbers of points, this fails. For example, the set of points in a plane, all added together, make a plane. Do you know about countability and the sizes of infinite sets? Would you like me to explain why one infinity can be bigger than another infinity?

That a circle can be thought of as the limit of a polygon as you increase the number of sides to infinity has been known for a long time, and this is how the Greeks (among others) first started to calculate pi. If you take a circle, and draw the smallest of a particular polygon (e.g. a triangle) that will fit around it (i.e. three tangents at 60°), and then the largest that will fit within it, the area of the circle must be between the areas of the two triangles. Now extend this to two squares, two regular pentagons, two regular hexagons....two regular Nagons...two identical circles.
It is not true, however, that a circle has infinite straight sides. A circle is "the locus of points a fixed distance [the radius] from a central point", and therefore is a curve.
Straight lines are curves, but they are special curves whose gradient (slope) is constant. A circle's slope cannot be defined except by a parameter or implicitly, but there are no corners (sudden changes of slope) anywhere along it so it is a single curve.

Thank's for your help on this. How, then, is a curve defined? I am familiar with the notion of some infinities being bigger than others, but I'd like to know more about 'countable unions', please.

I think that there are probably many definitions of a curve. One that might be useful here is the idea of a "connected" set of dimension 1. Connected means (roughly) that you can get from one point to another point without ever leaving the curve.

As to the countable unions business. Here is the distinction: The union (from n=1 to infinity) of the sets (1/n) is basically the set of all points which are 1/n for some integer, i.e. the set (1,0.5,0.333,0.25,...). This is a countable union, because there are the same number of sets as there are natural numbers (1,2,3...). However, let U_x =(x) (the set containing the real number x). Then the union over all x in the reals, is clearly all of the reals, this is because the number of sets is the same as the number of real numbers, which is uncountable. I've assumed you understand what countable and uncountable mean here (i.e. the size of the real numbers is bigger than the size of the natural numbers). Does this help, or is there anything still unclear?

Let's first see what a line consists of?

A line, be it straight line or curve ,consists of INFINITE number of points.
A line of length 2 meter has infinite points as also a line of length 3 meter or 10 meter or 100 lightyear.
Also, no FINITE number of points - however large the number be - can constitute a line.
In essence a line consists of infinite no. of points and nothing less than infinite no. of points.
So 2 points or 100 points or, say, 1 billion points cannot constitute a line.

More generally, infinite number of N dimensional entities can constitute an M(> N) dimensional entity. And, infinity of (N-2) dim. entities > infinity of (N-1) dim. entities > infinity of N dim. entities etc.
e.g. no.of points in a cube > no.of points on a square > no.of points on a linesegment ; EVEN IF in each case there are infinite no.of points .

Secondly; a line must have a length , a plane must have an area , a space must have a volume. There can be no line of length zero.

So now back to the "sides of circle" :
If a circle is to have infinite no.of sides then clearly its circumfarence will be infinite which is inadmissible.
Ofcourse circle is a limiting case of n-gon as n tends to infinity. And as long as n is finite the side is a straight linesegment. But when n is infinite the circle is a single piece object or it has one side as per Dan Goodman's definition.

In other words a curve and a straight line are two entirely different things and one cannot be thought of as a composition or a special case of the other.

Also, you can have an infinite number of sides (straight line segments of nonzero length) and still have a finite total length. For instance, let $P$ be the curve defined as follows: Draw a line segment from 0 at at angle of 90 degrees, of length 1. Continue this with a line segment at an angle of 45 degrees of length 14. Continue with a line segment of 22.5 degrees of length 1/9, and so on. Each of these line segments has length $1/n^2$ for some natural number $n$. The total length of this curve is then $\sum _{n=0}^{\infty }(1/n{)}^2=({\pi }^2)/6$ finite.