Re: I found a great FORMULA !!! pls check if corect !!!

From: Geert van der Wulp (GHFvanderWulp_at_hotmail.com)
Date: 09/22/04 

Date: Wed, 22 Sep 2004 14:04:39 +0000 (UTC)

On 22 Sep 2004, M. Suhaimi Ramly wrote:
>Hello all !!
>
>In high school we learn about a concept called 'diagonals' in polygons, which is basically lines that are connecting the dots around the polygon. We have this formula:
>
>For triangle (all sort, same length, or different side length) we have no diagonals = the sides dont count.
>
>For any square or rectangle (same length or different length) we can have two diagonals, like cross in the middle.
>
>For five, we have five, like a star. I dunno how to show without pictures :):):) but i hope you can imaginize.
>
>for six, we have nine (again, have to draw *wink* *wink* :))
>
>as my high school teacher used to say, i love to play around with formulas !! so i kind of find a pattern so here is the patern (quite complicated but it take days for me too :))
>
>the number of 'diagonals' is the number of sides multiplied with 0,1/2,1,1 1/2 (one and one half), 2, etc...let me explain how that works:
>
>like first 0x3=0 (diagonals in triangle) then 1/2x4=2 (diagonals in square), then 1x5=5 (diagonals in five side) and 1 1/2 x 6=9 !!! see it works perfectly :):)
>
>can you check if that formula is true ??
>
>suhaimi.

Hi Suhaimi,

Try to visualise this: Take a (regular!) polygon with n points. Then from each point we can draw N-3 lines to other points. This is because we don't draw a line from the point to itself, and the lines from the point to two neighbours are already drawn by the polygon.

This means: from each of the N points there will be N-3 lines: N * (N-3) lines in total. But of course we have counted each of the lines twice now. So the total number of diagonals is (1/2) * N * (N-3).

Your rule is: "the number of 'diagonals' is the number of sides multiplied with 0,1/2,1,1 1/2 ...". You will see that for a triangle this means: Number of sides * 0 = N * (1/2)*(3-3)
For a square: Number of sides * (1/2) = N * (1/2)*(4-3)

.......

Hope you see it.

With kind regards,

Geert van der Wulp

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