How many diagonals does a twenty sided polygon have?

2 Answers
smendyka
Apr 22, 2017

Explanation:

The number of diagonals of an n-sided polygon is:

#(n(n − 3))/ 2#

Substituting #20# for #n# gives:

#(20(20 - 3))/2 = 10(20 - 3) = 10 * 17 = 170#

NickTheTurtle
Apr 22, 2017

Any diagonal is drawn between two vertices. A 20-sided polygon has 20 vertices.

From Geogebra

The diagonals can be counted as #AB#, #AC#, #AD#, …, #BC#, #BD#, #BE#, …, #ST#, the combination of choosing #2# points from #20# vertices, or #""_20C_2=((20),(2))#. However, this method also counts the sides as diagonals. Remove the #20# sides to get #((20),(2))-20=190-20=170#.

In general, the diagonals for a convex #n#-sided polygon is #((n),(2))-n=(n!)/(2!*(n-2)!)-n=(n(n-1))/2-n=(n(n-3))/2#.

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