Which formula gives the number of diagonals in an n-sided polygon?

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Multiple Choice

Which formula gives the number of diagonals in an n-sided polygon?

Diagonals connect pairs of vertices that aren’t already connected by a side, so from a single vertex you can draw diagonals to all but itself and its two adjacent vertices. That means there are n − 3 diagonals emanating from each vertex. If you count this from every vertex, you get n(n − 3) diagonal-endings, but each diagonal has two ends, so you must divide by 2 to avoid double-counting. This yields n(n − 3)/2 as the total number of diagonals. For a quick check: a square has 2 diagonals, and the formula gives 4(1)/2 = 2; a triangle has 0 diagonals, and the formula gives 3(0)/2 = 0. The other expressions don’t account for counting each diagonal twice or they count all vertex pairs (including sides), so they don’t match the actual diagonal count.

Diagonals connect pairs of vertices that aren’t already connected by a side, so from a single vertex you can draw diagonals to all but itself and its two adjacent vertices. That means there are n − 3 diagonals emanating from each vertex. If you count this from every vertex, you get n(n − 3) diagonal-endings, but each diagonal has two ends, so you must divide by 2 to avoid double-counting. This yields n(n − 3)/2 as the total number of diagonals.

For a quick check: a square has 2 diagonals, and the formula gives 4(1)/2 = 2; a triangle has 0 diagonals, and the formula gives 3(0)/2 = 0. The other expressions don’t account for counting each diagonal twice or they count all vertex pairs (including sides), so they don’t match the actual diagonal count.

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