How many diagonals does a regular octagon have?


 Jun 26, 2020

If you are standing at any vertex of a convex polygon, there are three vertices to which you cannot draw a diagonal.

You cannot draw a diagonal to the vertex on your left (it would be a side, not a diagonal); you cannot draw a

diagonal to the vertex that you are standing on; and you cannot draw a diagonal to the vertex to your right

(it also would be a side). 


You can draw diagonals to each of the other vertices. So, if there are n-vertices, you can draw n - 3 diagonals.

This will be true for each of the vertices:  n·(n - 3).

However, when you draw a diagonal from vertex A to vertex G, this will be redrawn as a diagonal from 

vertex G to vertex A; so, we'll have to divide the expression above by 2.


The final formula for the number of diagonals in a convex n-gon is:  n·(n - 3) / 2


For an octagon, it will be:  8·(8 - 3) / 2

 Jun 26, 2020

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