   SEARCH HOME Math Central Quandaries & Queries  Question from Megan: OK, so I've been trying to figure this one out for a few days and I keep going around in circles... or maybe octagons. I'm trying to develop a relationship between the number of points of a regular polygon and the (a) number of lines you could draw between those points, (b) number of triangles you could draw, (c) number of quadrilaterals, etc. Lines was relatively simple. 2 points = 1 line = 1 3 points = 3 lines (+2) = 1+2 4 points = 6 lines (+3) = 1+2+3 5 points = 10 lines (+4) = 1+2+3+4 etc. So if n is the number of sides/vertices, and L is number of lines, we have: L = (n)(n-1) / 2 So I looked at triangles and I noticed a similar pattern: 3 points = 1 triangle = 1 4 points = 4 triangles (+3) = 1 + (1+2) 5 points = 10 triangles (+6) = 1 + (1+2) + (1+2+3) 6 points = 20 triangles (+10) = 1 + (1+2) + (1+2+3) + (1+2+3+4) ... but I can't figure out an actual equation for this (or for the quads thing) Please help Hi Megan,

Let's first look at lines in a polygon with n vertices. If you are going to join two vertices there are n choices for the starting point. Once you have the starting point there are n-1 choices for the end point. Thus it seems you have n(n - 1) possible lines, but you have counted each line twice, once in each direction. Thus the number of lines joining vertices is n(n - 1)/2, exactly what you got.

Now look at triangles. To draw a triangle you first choose a starting point and you have n possible choices. There are then n - 1 choices for the second point and n - 2 choices for the third point. Hence, as for lines, it seems you have n(n - 1)(n - 2) possible triangles. But consider a particular triangle. How many times did you count it? There are 3 way to choose one of its vertices as the first vertex, 2 ways to choose the second vertex and only 1 way to choose the third vertex. Thus each triangle got counted 3 × 2 × 1 times. Hence the triangles joining vertices of a polygon with n vertices is n(n - 1)(n - 2)/(3 × 2 × 1).

I hope this helps,
Penny     Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.