How many diagonals does a twenty-sided polygon have?

Question

A convex polygon has 20 sides. How many diagonals does it have? Find the general formula for n sides and explain it in two different ways, then use the same style of counting to say how many points inside the polygon are crossings of two diagonals, assuming no three diagonals meet at a point.

Accepted Answer

170 diagonals. In general, n(n minus 3)/2. Method 1: pairs of corners. Any two of the 20 corners are joined by a line segment: 20 choose 2 = 190 segments. Twenty of these are the sides of the polygon; the rest are diagonals: 190 minus 20 = 170. Method 2: corner by corner. From one corner, you can draw a diagonal to every corner except itself and its two neighbours: 20 minus 3 = 17 diagonals. Twenty corners give 20 × 17 = 340, but each diagonal has been counted from both of its ends, so divide by 2: 170. The formula. n(n minus 3)/2: a triangle has 0, a square 2, a pentagon 5, a hexagon 9, a de…

How many diagonals does a twenty-sided polygon have?

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