# How many triangles can be formed from N points on a circle?

I have a circle with N points on it, and I want to determine how many triangles can be formed using these points.

How can I do this?

Thanks!

Andrew

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The vertices of those triangles have to be from these $N$ points, right? –  Patrick Li Oct 3 '12 at 20:24
@PatrickLi Yes - that is right. –  AndyGeek Oct 3 '12 at 20:25

Each set of $3$ of the $N$ points determines a triangle, and each triangle is determined in this way, so all you have to do is determine how many $3$-element subsets a set of $N$ things has. If you don’t already know this, you should read this article.
That's one way of interpreting the question. Another (more interesting, in my opinion) is to look at all triangles that can be generated by chords using $N$ points on a circle. For example, with $N=4$ points we have, essentially, divided the circle and its interior into 4 triangular regions, defined by a square and its two diagonals. For $N=5$ we'll have 11 possible triangles (look at $K_5$). If you allow overlapping triangles, you get an even larger number. –  Rick Decker Oct 4 '12 at 0:41