Diagonals of a 12-gon

Question

How many diagonals does a convex polygon with 12 vertices have?

Accepted Answer

A diagonal is an unordered pair of vertices that is not an edge. Count all pairs, subtract the 12 sides: binom122 - 12 = 66 - 12 = 54. Approach 2, per vertex Each vertex sends diagonals to the 12 - 3 = 9 vertices that aren't itself or its two neighbors: 12 times 9 = 108 directed diagonals, and each diagonal was counted from both ends: frac12 times 92 = 54. The general formula is fracn(n-3)2, but the derivation is the deliverable, not the formula. The classic miss is forgetting the divide-by-2 (a diagonal has two endpoints). Sanity-check on a square: frac4 cdot 12 = 2 diagonals, correct. Checking tiny cases before announcing a formula is exactly the habit speed-round interviewers grade.

Diagonals of a 12-gon

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