23.3.1. Finding Number of Diagonals

The third question is how many diagonals are there in a convex polygon with n sides? So, again we can derive the formula using induction or some other mechanism but we will count it directly.

This chunk introduces the problem of finding the number of diagonals in a convex polygon with 'n' sides. A diagonal connects two non-adjacent vertices, and since the polygon is convex, we can count the diagonals directly by considering the properties of the vertices.

Think of a convex polygon as a flat piece of string bent into a shape, like a triangle or a pentagon. The vertices are points along this string. Counting the diagonals is like counting the rope sections that stretch across the inside of the shape without touching the edges.

Imagine you are given a convex polygon consisting of n sides and where the vertices are v1 to vn, now let us focus on some arbitrary vertex vi and try to count the number of diagonals that we can have where vi is one of the end points.

We start by selecting an arbitrary vertex, say 'vi'. To form a diagonal with 'vi', we cannot connect it to its immediate neighbors 'vi-1' and 'vi+1', nor can we connect it to itself. Hence, the valid choices for the other end of the diagonal exclude these three vertices. This results in (n - 3) possible diagonals for each vertex.

Imagine standing at one corner of a square room (your vertex). You can't shake hands (make a diagonal) with the people next to you (immediate neighbors), nor can you shake hands with yourself. The number of people left to shake hands with is like the remaining vertices you can connect with.

So, now it turns out that the total number of diagonals will be n(n-3)/2. Why over 2? Because what we did here is with vi being one of the end points I have n - 3 diagonals.

The final formula n(n-3)/2 calculates the total number of diagonals. We've counted diagonals for each vertex vi, resulting in n(n-3) counts. However, this counts each diagonal twice (once for each endpoint). To get the actual number, we divide by 2.

Consider each handshake you count when you say hello to people in a party. You count every handshake when you say hi to your friend, and your friend counts the same when they greet you. To know how many unique handshakes happened, you divide by 2!

But now my k can range from 1 to n; my k could be vertex number v1, my k could be vertex number v2 and so on. So, if I take the summation over k being equal to 1 to n then I get the total number of triangulations for n + 2 sided convex polygon.

The total diagonal count formula gives us a comprehensive understanding. It highlights that diagonals connect every vertex except immediate neighbors, demonstrating how each has (n - 3) options. This provides a consistent method for counting across all vertices.

Think of a group project where each student (vertex) must connect with every other student (make diagonals) but can’t link with their direct partners (neighbors). Summing up all potential connections gives insight into collaboration opportunities among the students.

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