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11 sides

The diagonal of a polygon is a line segment that joins two non adjacent vertices. So for a n-sided polygon, each vertex can be joined to n-3 other sides. So there ought to be n x (n-3) diagonals for a n-sided polygon; however, each diagonal is counted twice, once by the vertex at each end. So the number of diagonals is:

diagonals = n (n-3) / 2

So for 44 diagonals:

44 = n (n-3) / 2

88 = n2 - 3n

n2 - 3n - 88 = 0

(n - 11)(n + 8) = 0

n = 11 or -8

Since the number of sides can't be negative, the number of sides is 11, a hendecagon (or undecagon)

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Q: What is the number of sides of polygons which has 44diagonals?
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