How many diagonals does an octagon (a polygon with 8 sides) have?
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → 20
Joining any 2 corners of a polygon gives a side or a diagonal, so diagonals \(=\,^nC_2-n\).
Lines joining 2 of the 8 corners: \(\displaystyle ^8C_2=\frac{8\times7}{2}=28\)
8 of these are sides, so diagonals \(=28-8=20\)
Why the other options are incorrect
Option (2) 28 — Incorrect. This is \(^8C_2\); it still includes the 8 sides.
Option (3) 16 — Incorrect. This is \(2\times8\), which is not a counting rule.
Option (4) 40 — Incorrect. This is \(8\times5\): from each corner 5 diagonals start, but each diagonal has two ends, so it must be halved.