algebra and trigonometry

Question

find the number of sides in a polygon whose interior angles are in arithmetical progression. the smallest being 120° and the common difference 5 °

Answer 1

I like visual solutions to problems like this.

In 1967, a team at MIT developed “Turtle Graphics” and the Logo programming language to teach very simple programming to children.  The “turtle” moved across the floor as directed by the programming commands.  There are interactive versions of Logo (to download) and some run from web hosts, e.g. –

         http://turtleacademy.com/registration.php

 

If a “turtle” began at a certain spot, pointing in a certain direction, and proceeded to draw a line, move, turn left, move, turn left, move, …, and finally crossing the initial line, the next turn would make a total of 360 degrees of left turns.  This is because the sum of the external angles of a polygon equals 360.  Depending on the number of sides (N) of the polygon, this makes the sum of the internal angles 180(N-2) degrees.  [I think it is a lot more fun to see than to memorize this.]

 

This problem describes an arithmetic progression of the size of the “left turns” required to make an n-sided, irregular polygon.  The sides (which also must be unequal, or else you create a spiral) are not described, but must be counted in order to answer the question. 

 

The progression is:   120, 125, 130, 135, … . The common difference is 5 degrees.

 

So, mathematically, what the problem is really asking is, “When will the sum of the external angles equal 360 degrees for an arithmetic progression of exterior angles of 60, 55, 50, 45, …? (with a common difference of -5 degrees)?”  If you can translate the problem, you are close to solving it.

 

Now, the Sum of an arithmetic sequence is (N/2)(2A + (N-1)D)  

              Where A=first term

                          D = common difference

                          N = number of terms

 

So, what is N when this sum is equal to 360 degrees?

                      (N/2)( 2(60) + (N-1)(-5) ) = 360

                           60N – (5/2)N^2 + (5/2)N  = 360     [expand terms]

                             -24N  + N^2  - N = -144               [multiply everything by -2/5]

                             N^2 – 25N + 144 = 0                   [ rearrange and collect terms]

                  N= 9  or  N= -16     [Now, after N=13 the expression goes negative]

                            N=9 sides

 

The sum of the external angles is 360 degrees.  The sum of the interior angles is

       (9-2)*180=1260.

 

O.K. let’s go see it in action (at the link above).  Type in (or copy/paste):

 

   ClearScreen

   For [A 60 20 -5] [FORWARD :A*5*(20/60)  LEFT :A]

 

Final Note:  To properly adjust the length of each side, you will have to learn trigonometry; but that’s really fun with graphics, color, animation, and sound!

Answer 2

The formula for the sum of the interior angles of a polygon is:

S = (n-2)180 where n is the number of sides

 

This means the sum of the interior angles will be a multiple of 180

 

The multiples of 180 are... 180, 360, 540, 720, 900, 1080, 1260, ...

 

Let's look at the progression of the totals of our angles on this polygon

 

Angle #     Angle size    Total of Angles   Total=mult

                                                          of 180 *Y or N*

------------------------------------------------------------

  1             120               120                      No

  2             125               245                      No

  3             130               375                      No

  4             135               510                      No

  5             140               650                      No

  6             145               795                      No

  7             150               945                      No

  8             155              1100                     N0

  9             160              1260                    YES

 

Since 9 interior angles in this progression gives us a multiple of 180

Then there are 9 interior angles in the polygon

 

The formula says 1260 = 180(n-2)

                          1260/180 = n-2

                          7=n-2  

                          n=9

 

There are 9 sides to the polygon