## 44 or 45 Diagonals

What is the number of sides of a regular polygon for which the number of diagonals is $44$ or $45$?
Source: NCTM Mathematics Teacher 2006

SOLUTION
The number of diagonals of a convex polygon of $n$ vertices (or sides) is given by $n(n-3)/2$. Please refer to post titled “Convex Polygon” posted on 6/18/2011 for details.
Case 1: $44$ diagonals
$n(n-3)/2=44$
$n(n-3)=88$
Let’s try a few numbers
$7(4)=28$
$8(5)=40$
$9(6)=54$
$10(7)=70$
$11(8)=88$
The polygon has $11$ sides.
Case 2: $45$ diagonals
$n(n-3)=90$
Continuing where we left off above we get
$12(9)=108$
$13(10)=130$
$\cdots$
Clearly, there is no polygon that has $45$ diagonals.

Answer: $11$