23 gon polygon has 230 diagonals

Solution:

The number of diagonals in polygon is given as:

where n is the number of polygon sides

Given that,

Consider a regular 23-gon

n = 23

Substitute n = 23 in above formula

Thus, a 23 gon polygon has 230 diagonals

90

=====================================

Work Shown:

n = number of vertices = 15

d = number of diagonals

So there are 90 diagonals for this 15-gon.

The numerator n(n-3) is from the fact that there are n vertices, and for each vertex, there are n-3 points to connect to that will form a diagonal. Why n-3? The 3 comes from the fact that you cannot select the current point, or either neighboring point (on either side). So we ignore 3 points when forming diagonals for any vertex. That's why n drops to n-3.

The 2 in the denominator is to correct for double-counting. A diagonal like AB is the same as BA. The order does not matter when it comes to naming a segment.

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