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Diagonals of a Polygon
From Greek: dia- "across" + -gonia "angle,"
Definition: The diagonal of a polygon is a
line segment
linking two non-adjacent
vertices.
Try this Adjust the number of sides of the polygon below, or drag a vertex
to note the behavior of the diagonals.
(If there is no image below, see support page.)
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 the the number of vertices or sides, or (n-3).
There are N vertices, which gives us n(n-3) 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:
N is the number of sides (or vertices) 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 topicsGeneral
Angles in a circleArcs
(C) 2007 Copyright John Page
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