How many diagonals does an octagon have?

How many diagonals does an octagon have?

• A. 12
• B. 18
• C. 20
• D. 24

Answer: (C)

Think about how many diagonals connect to the octagon’s vertices. From any given vertex, you can connect to all vertices except itself and its two adjacent neighbors, leaving 8 − 3 = 5 diagonals per vertex. Counting this for every vertex gives 8 × 5 = 40 diagonal instances, but each diagonal is counted twice (once from each end), so you divide by 2 to get the total: 40 / 2 = 20. Another way to see it: there are C(8, 2) = 28 pairs of vertices in total. Of these, 8 pairs are sides, leaving 28 − 8 = 20 diagonals. So the octagon has 20 diagonals.

Think about how many diagonals connect to the octagon’s vertices. From any given vertex, you can connect to all vertices except itself and its two adjacent neighbors, leaving 8 − 3 = 5 diagonals per vertex. Counting this for every vertex gives 8 × 5 = 40 diagonal instances, but each diagonal is counted twice (once from each end), so you divide by 2 to get the total: 40 / 2 = 20.

Another way to see it: there are C(8, 2) = 28 pairs of vertices in total. Of these, 8 pairs are sides, leaving 28 − 8 = 20 diagonals.

So the octagon has 20 diagonals.