Diagonals of Large Polygons

Geometry Level 3

We know that a diagonal is a segment who joins two non adjacent vertices. By looking at the above figure, we notice that a square has two diagonals, a pentagon has five diagonals, a hexagon has nine diagonals...etc.

Suppose we have a polygon with 2017 sides. What would be the number of its diagonals?

2017000 2021119 2000000 2031119 2016000

1 solution

Chew-Seong Cheong
Jul 28, 2017

We note that a $n$ -sided polygon has $n$ vertices. It takes two vertices to produce a line. Therefore, the total number of lines produced is $\displaystyle {n \choose 2} = \dfrac {n(n-1)}2$ . Since all lines produced other than the $n$ sides are diagonals, the number of diagonals is $n_d = \dfrac {n(n-1)}2 - n = \dfrac {n(n-3)}2$ . When $n=2017 \implies n_d = \dfrac {2017\times 2014}2 = \boxed{2031119}$ .

Thank you. I solved it the same way too. $2017 \choose 2$ $- 2017 = 2031119$ .

Hana Wehbi - 3 years, 10 months ago

Hana, it is Brilliant.org standard if not LaTex standard that numbers on their own need not to be in LaTex. I have been editing your problem.

Chew-Seong Cheong - 3 years, 10 months ago

Ok, l will edit them. Sometimes l feel lazy to use Latex :)

Hana Wehbi - 3 years, 10 months ago

@Hana Wehbi – It is okay. Just don't include them in your future.

Chew-Seong Cheong - 3 years, 10 months ago

@Chew-Seong Cheong – I won't. I promise.

Hana Wehbi - 3 years, 10 months ago

Diagonals of Large Polygons

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