The given number of diagonals of the polygon.

Geometry Level 2

A polygon has a total number of given diagonals.

It is 65 \displaystyle \boxed { 65 } .

How many sides does the polygon have?


The answer is 13.

. . by . . , 14, South Korea

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1 solution

. .
Feb 25, 2021

The diagonal formula for n-gon is n ( n − 3 ) 2 \displaystyle \frac { n ( n - 3 ) } { 2 } .

Then n ( n − 3 ) 2 = 65 → n ( n − 3 ) = 130 → n 2 − 3 n − 130 = 0 → ( n − 13 ) ( n + 10 ) = 0 → n = 13 , n = − 10 \displaystyle \frac { n ( n - 3 ) } { 2 } = 65 \rightarrow n ( n - 3 ) = 130 \rightarrow n ^ { 2 } - 3n - 130 = 0 \rightarrow ( n - 13 ) ( n + 10 ) = 0 \rightarrow n = 13, n = -10 .

But the number of diagonals cannot be negative, so it is 13 \boxed { 13 } .

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