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 polygonnumber of
diagonals | |
30 | |
42 | |
55 | |
69 | |
-
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.