How is the number of diagonals in a polygon with n sides calculated?

To find the number of diagonals in a polygon with ( n ) sides, the correct formula is based on the understanding of how many connections can be made between non-adjacent vertices.

In a polygon, each vertex can connect to ( n - 1 ) other vertices. However, diagonals are connections that do not include the edges of the polygon. Since each vertex connects to two adjacent vertices (which form the sides of the polygon), we need to subtract these two connections. Thus, each vertex can connect to ( n - 3 ) vertices through diagonals.

Since there are ( n ) vertices in the polygon, the total number of diagonal connections counted will initially seem to be ( n(n - 3) ). However, this counts each diagonal twice (once from each endpoint), so you must divide by 2 to get the actual number of unique diagonals.

Therefore, the formula for the number of diagonals in a polygon with ( n ) sides is:

[

\frac{n(n - 3)}{2}

]

This is why the correct answer is based on this calculation method, providing a clear pathway from the number of vertices to the counting of non-adjacent connections