# Thread: Spheres and circles and things

1. ## Spheres and circles and things

Hey everyone,

I was just curious about something:

Say you had a normal polygon with $n$ sides. As $n$ approaches infinity, your polygon begins to look more and more like a circle.

We could define a straight line to be 180 degrees.

We know that the equation of the sum of the interior angles $\theta i$ of a polygon is:

$\theta i = (n - 2)(180) = 180n - 360$

We know that the measure of a single interior angle $\alpha$ is:

$\alpha = \frac {180n - 360}{n}$

When we take the limit of n as n approaches zero:

$\lim_{n\rightarrow\infty} \frac {180n - 360}{n} = 180$ By L'Hopitals rule

It's weird to me, because it seems like the curve of a circle is flat, but also curved. What if you visualized this in three dimensions? Does that mean that the surface of a sphere is 2 dimensional?

2. Hi,

actually, the same 180° value would have appeared with any smooth curve, not specifically with a circle. The reason for this is that if you draw a line between a fixed point A on a curve and another point M on the same curve then, as the point M tends to A along the curve, the line (AM) converges (in a sense that can be made rigorous) to the tangent line to the curve at A. This tangent line is the same if M tends to A from the left or the right, hence the flat angle (the angle between the two directions of the tangent at A).

In three dimensions, on a smooth surface, you have a similar property, namely that the line (AM) would get nearer and nearer to the tangent plane as M tends to A, and a plane (AMN) would converge to the tangent plane as M and N tend to A along curves drawn on the surface that are not tangent to each other at A.