Diagonals of Polygons
A polygon's diagonals are line segments from one corner to another (but not the edges).
A square has
2 diagonals
An octagon has
20 diagonals
The number of diagonals of an n-sided polygon is:
n(n − 3) / 2
Why does this work?
- Each of the n corners joins to (n − 1) other corners
- Two of those are edges, so there are (n − 3) diagonals from each corner
- That seems like n(n − 3) diagonals, but each diagonal is counted twice (once from each end),
- so we divide by 2: n(n − 3) / 2
Examples:
- a square (or any quadrilateral) has 4(4−3)/2 = 4×1/2 = 2 diagonals
- an octagon has 8(8−3)/2 = 8×5/2 = 20 diagonals
- a triangle has 3(3−3)/2 = 3×0/2 = 0 diagonals
Try it Yourself:
images/area-coords.js?mode=diag
A diagonal can actually be outside the polygon,
which happens with some concave polygons
Handy Table
Number of diagonals for the first few polygons:
| Polygon | Sides (n) | Diagonals |
|---|---|---|
| Triangle | 3 | 0 |
| Quadrilateral | 4 | 2 |
| Pentagon | 5 | 5 |
| Hexagon | 6 | 9 |
| Heptagon | 7 | 14 |
| Octagon | 8 | 20 |
| Nonagon | 9 | 27 |
| Decagon | 10 | 35 |
1794, 7625, 7627, 7629, 11, 1795, 7626, 7628, 7630, 7631