Counting diagonals, step by step
A twenty-sided shape has 170 diagonals.
The question
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
- Start from the only number given: 14 of them, arranged so each has two neighbors. Nothing is counted yet.
- Join EVERY pair of the 14 first, the ones already joined included. Each corner reaches 13 others, so that is 14 × 13 = 182 — and every line got counted from both ends, so halve it: 182 ÷ 2 = 91.
- That 91 is a real count and it is not the answer: 14 of them are its sides, joining each corner to its two neighbors, and those were never the thing being counted. There are exactly 14 of them, one per corner.
- Take them off and what is left is the diagonals that cut across: 91 − 14 = 77.
How it works.
- 01
Start from the only number given: 14 of them, arranged so each has two neighbors. Nothing is counted yet.
- 02
Join EVERY pair of the 14 first, the ones already joined included. Each corner reaches 13 others, so that is 14 × 13 = 182 — and every line got counted from both ends, so halve it: 182 ÷ 2 = 91.
- 03
That 91 is a real count and it is not the answer: 14 of them are its sides, joining each corner to its two neighbors, and those were never the thing being counted. There are exactly 14 of them, one per corner.
- 04
Take them off and what is left is the diagonals that cut across: 91 − 14 = 77.