Hints and solutions to
Problem of the day for Thursday, November 7

The number of diagonals of a regular polygon is subtracted from the number of sides of the polygon and the result is zero. What is the number of sides of this polygon?


  • By drawing a few pictures, you can make the following table:

    sides of polygon

    number of diagonals

    3

    0

    4

    2

    5

    5

    6

    9

  • So it appears the answer is the pentagon (5 sides). Can we come up with a reason for this?
  • How many diagonals toes a regular polygon have in general? Well, from each vertex of the polygon, you can draw diagonals to all the other points except the point itself and the two neighboring points. If the polygon has n sides (and hence n vertices), then each vertex is connected to n-3 other vertices by diagaonals.
  • This makes n(n-3) all together. But in doing this, we've counted each diagonal twice (once at each end). So the polygon with n sides has n(n-3)/2 diagonals. This is equal to n precisely when n=5.