How many diagonals can be drawn by joining the vertices of an octagon?

Answer: (a) 20

Any two of the 8 vertices can be joined in C(8, 2) = 28 ways. Eight of those joins are the sides, so 20 are diagonals.

1. Lines joining two vertices: C(8, 2) = 8 × 7 ÷ 2 = 28.
2. Of these, 8 are sides of the octagon.
3. Diagonals = 28 − 8 = 20.
4. Check with the formula n(n − 3)/2 = 8 × 5 ÷ 2 = 20.

Remember · Diagonals of an n-sided polygon = n(n − 3)/2: each vertex joins n − 3 non-adjacent vertices, and each diagonal is counted twice.

Question and answer: UPSC's official GS Paper II (2018, Series A) — paper ↗ · answer key ↗. Explanation: Minimalist IAS, checked 30 Sept 2026 (how we verify). · Report an error