A diagonal is a line segment connecting two non-adjacent vertices of a polygon. There is a neat formula to calculate the number of diagonals in any polygon.
A triangle has 0 diagonals. A quadrilateral has exactly 2 diagonals. These are useful facts to verify the formula.
Number of Diagonals = n(n-3) / 2
Where n = number of sides (or vertices) of the polygon.
| Polygon | Sides (n) | Formula | Diagonals |
|---|---|---|---|
| Triangle | 3 | 3(3-3)/2 | 0 |
| Quadrilateral | 4 | 4(4-3)/2 | 2 |
| Pentagon | 5 | 5(5-3)/2 | 5 |
| Hexagon | 6 | 6(6-3)/2 | 9 |
| Octagon | 8 | 8(8-3)/2 | 20 |
| Decagon | 10 | 10(10-3)/2 | 35 |
From any vertex, you can draw diagonals to (n-3) other vertices (you cannot draw to itself or its 2 adjacent vertices).
Total = n × (n-3) — but each diagonal is counted twice (once from each end).
So divide by 2: n(n-3)/2
n = 6: Diagonals = 6(6-3)/2 = 6×3/2 = **9 diagonals**
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