
Diagonals of a Polygon
From Greek: dia "across" + gonia "angle,"
Try this Adjust the number of sides of the polygon below, or drag a vertex
to note the behavior of the diagonals.
A diagonal of a polygon is a line segment
joining two vertices.
From any given vertex, there is no diagonal to the vertex on either side of it,
since that would lay on top of a side.
Also, there is obviously no diagonal from a vertex back to itself.
This means there are three less diagonals than there are vertices.
(diagonals to itself and one either side are not counted).
Formula for the number of diagonals
As described above, the number of diagonals from a single vertex is three less than the the number of vertices or sides, or (n3).
There are N vertices, which gives us n(n3) diagonals
But each diagonal has two ends, so this would count each one twice. So as a final step we divide by 2, for the final formula:
where
N is the number of sides (or vertices)
Concave polygons
One of the characteristics of a concave polygon is that some diagonals will lie outside the polygon. In the figure above
uncheck the 'regular" checkbox. The drag one of the vertices towards the center of the polygon. You will see white areas appear.
The polygon is filled with a yellow color, so what you are seeing is a diagonal that lies outside the concave polygon.
An easy mistake
The above formula gives us the number of distinct diagonals  that is, the number of actual line segments.
It is easy to miscount the diagonals of a polygon when doing it by eye.
If you glance quickly at the
pentagon on the right, you may be tempted to
say that the number of diagonals is 10. After all, there are two at each vertex, and 5 vertices.
Some people see them making three triangles, for 6 diagonals.
But there are only 5 diagonals. Count them carefully.
Other circle topics
General
Equations of a circle
Angles in a circle
Arcs
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