Counting diagonals
The question
A polygon has 14 sides.
How many diagonals does it have?
The same count, minus the ones already joined
corners:14
14×13=182
182÷2=91
all lines:91· already joined:14
91−14=77
- There are 14 of them, and each already has two neighbors.
- First join every pair of the 14, neighbors included. Each corner reaches 13 others, so that is 14 × 13 = 182 — but each line was counted from both ends, so halve it: 182 ÷ 2 = 91.
- But 14 of them are its sides, joining each corner to its neighbors, and those are not being asked about.
- Take them off to leave the diagonals that cut across: 91 − 14 = 77.
How it works.
- 01
There are 14 of them, and each already has two neighbors.
- 02
First join every pair of the 14, neighbors included. Each corner reaches 13 others, so that is 14 × 13 = 182 — but each line was counted from both ends, so halve it: 182 ÷ 2 = 91.
- 03
But 14 of them are its sides, joining each corner to its neighbors, and those are not being asked about.
- 04
Take them off to leave the diagonals that cut across: 91 − 14 = 77.