find the number of all possible diagonals in a polygon a. 40 sides
Answers
Answer:
Does a 40 sided polygon have 6840 diagonals? The formula employed to find the numbers of diagonals in a polygon isn ( n-3 ) /2 . 30 x 27/2 = 810/2 = 405 diagonals .
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Answer:
The question apparently is "How many diagonals does a polygon with n
sides have?"
You have remembered the first formula correctly: it is n(n-3)/2.
One way to see this is to notice that you can draw (n-3) diagonals
from every vertex of the polygon. This is because there are (n-1)
other vertices, but two of them are adjacent vertices and so
don't count towards making diagonals. This seems to give n(n-2)
diagonals, but this way of counting counts every diagonal twice since
each diagonal connects to two vertices, so you have to divide by 2.
By a recursive formula, we mean a way of expressing the answer for n
vertices in terms of the answer for n-1 vertices. Suppose, for
example, that you already know the answer for a polygon with n-1
vertices. Now if you add another vertex between two of the vertices
of the original polygon, then all the diagonals of the original
polygon will still be diagonals of the new polygon, and so will the
side joining the two vertices that you added a new vertex between, and
so will the line segments joining the new vertex to all the other
vertices of the original polygon. (Got all that?) So if we let diag(n)
be the number of diagonals for a polygon with n sides, we get the
formula:
diag(n) = diag(n-1) + n - 3 + 1 or
diag(n-1) + n - 2
The first formula is better, since it actually gives you the answer.
But sometimes it's easier to get a recursive formula first and use
that to get an explicit formula (your first formula is an explicit one
since you only need the number of vertices in the polygon to get the
number of diagonals in that polygon). This is called "solving the
recursion." Sometimes a recursive formula is the best you can do
because there simply is no explicit formula.