The expression for the number of diagonals that we can make from one vertex of…

A diagonal of a polygon joins two vertices that are not already joined by a side (i.e., they are not adjacent). From any single vertex of an n-sided polygon, straight lines can be drawn to each of the remaining (n − 1) vertices; of these connections, exactly 2 go to the vertices immediately next to the chosen vertex, and those two connections are the polygon's own sides, not diagonals.

1. An n-sided polygon has n vertices in total.

2. From one chosen vertex, straight lines can be drawn to (n − 1) other vertices — every vertex except itself.

3. Of these (n − 1) vertices, 2 are directly adjacent to the chosen vertex; the segments to them are the polygon’s own sides, so they must be excluded from the diagonal count.

4. The number of diagonals from that one vertex is therefore (n − 1) − 2 = n − 3.

As a check, take a pentagon (n = 5), where the formula gives 5 − 3 = 2. Labelling its vertices A, B, C, D, E in order, the sides from A go to B and E, so those two are excluded; the only vertices left to join by a diagonal are C and D — exactly 2, confirming the n − 3 expression.