In a regular octagon, how many diagonals are possible?

Concept: For any convex polygon with n vertices, every pair of vertices is joined either by a side or by a diagonal. So the number of diagonals D = (ways to choose 2 vertices) minus (number of sides) = nC2 − n, which simplifies to n(n − 3)/2.

Application: Apply this to an octagon, where n = 8.

1. Total ways to choose 2 vertices out of 8: 8C2 = (8 × 7) / 2 = 28. Each such pair joins two vertices — either along a side of the octagon or as a diagonal.

2. An octagon has 8 sides, and each side is one such vertex-pair that is NOT a diagonal, so subtract these: 28 − 8 = 20.

Cross-check: Using the simplified form n(n − 3)/2 directly: 8 × (8 − 3)/2 = 8 × 5 / 2 = 20 — the same result confirms the calculation.

So a regular octagon has 20 diagonals.