Polygon Diagonal Thorem?

Question

The mathematical definition of a diagonal is "an adjacent line that bisects a polygon." Does anyone know of an theorem/equation that will determine how many diagonals a polygon has, without having to do a drawing to see how many there are? An example is how many diagonals would a decagon (10 sided polygon) have?

Answer 1

The equation for the number of diagonals in any regular polygon is:

n(n-3)/2, where n is the number of sides in the polygon

Answer 2

The diagonal of a polygon is any line which joins two vertices of the polygon but is not the side of the polygon.The formula is :
The number of diagonals, d (n),of a polygon with n vertices
d(n) = C(n,2) -n = (1/2) n(n - 3) Answer.
For n=3, d(3) = 0 OK for triangle.
For n=4, d(4) = 2 OK for quadrilateral.
For n=5, d(5) = 5 OK for triangle.
For n=10, d(10) = 35 for a decagon..

Answer 3

I think an n-sided polygon has n(n-3)/2 diagonals, so for a 10-sided polygon it would be 35.

Here is my reasoning. For an n-sided polygon, there are n-3 diagonals from each vertex (one to every other vertex that is not adjacent). Notice however, that this counts the number of diagonals twice, since every diagonal touches precisely two vertices. That the number of diagonals is n(n-3)/2

Answer 4

If you don't count the sides of the polygon, for an n-sided polygon, n-2 lines can be drawn from a vertice to n-3 other vertices. That's n(n-3) in all. But since that's counting the same lines twice, the correct formula is

N = 1/2 n (n-3)

For example an hexagon has 1/2 6 (3) = 9

Answer 5

for polygons of an even number of sides, the number of diagonals that bisect the polygon is 1/2 the number of sides. Polygons with an odd number of sides have no lines between vertices which bisect them, but you can draw n altitudes which bisect them, n being the number of sides.

Answer 6

The number of bisections would be the same as the number of sides.
Take a triangle, for instance: you can draw a line from each corner to the opposite edge for each of the three corners. For regular polygons with an even number of sides, like a square, you could draw a line from corner to corner (twice, for a square; five times for a decagon) and edge to edge (the same number of times).

Of course a line can be drawn through a square or other regular even-number-sided polygon in infinitely many ways to create two congruent halves- go ahead, try it.

Are you certain you have the correct definition?

Answer 7

Yes the polygon with diagonals equal to 65 exists. It will have 13 sides. n(n-3)/2=65. n^2-3n-130=0 so n=13 and n= -10. In the same way u can chk it for 80. I t wont exist.

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Answer 8

Ok, if a polygon has 10 sides, it can be bi-sected 5 times.
(bisected, or cut in half)
How about
d=s/2
where d=diagonals
s=sides