How many sides does a regular polygon have?

Question

The interior angles of a regular polygon each measure 165 degrees.  How many sides does the polygon have? Please explain how you arrived at your answer.  Thank you

Answer 1

Janet, 

 

Here is an easy way to solve the problem:

 

1) The interior and the exterior angles

    make a linear pair: add up to make 180°.

 

     180° - 165° = 15° measure of one ext. angle

 

2) the sum of the exterior angles of any convex polygon is 360°

 

       360° ÷ 15° = 24 exterior angles

 

3) the number of exterior angles = the number of interior angles = the number of sides in any convex polygon

 

4) Therefor there are 24 sides in this regular polygon

 

 

Answer 2

In the late 1960’s, Seymour Papert and a team at MIT developed “Turtle Graphics” and the Logo Programming Language to teach kids how to program a computer.  A programmable “turtle” was placed on the floor and given instructions.  To use Logo for free on-line, register at:
   http://www.logointerpreter.com
 
Select “Editor” and enter the following program as new logo commands.  Execute it (click the rocket blasting off). 

       repeat 24 [
          forward 30
          left 15
       ]

 

 

To dray a "regular polygon," the Turtle moves the same amount for each side and turns the same amount each time.  When the Turtle reaches its starting point, it has turned completely around.  That is an extremely important principle in geometry:

  The sum of the exterior angles of a regular polygon is 360°.

 

Ah, but problem asks you to find the number of sides (that is also the number of angles) when you are given the interior angle.

 

Let n = the number of sides; also the number of angles

 

Now, realize that an interior angle = (180° - the exterior angle)   (that is, they are supplementary).

 

Sum of interior angles = n*(a single interior angle)

                                = n*(180˜ - a single exterior angle)

                                = 180°n - n*(a single exterior angle)

                                 = 180°n - 360°

 Sum of interior angles = 180°(n-2)

 

This is really easy !  

 

For

      n*165° = 180°(n-2)     [Sum of interior angles = n*(a single interior angle)]

     165n = 180n - 360

    -15n = -360

      n = 24

                                                     PLZ contact me for on-line tutoring.

Answer 3

Hello Janet,

 

You could solve this problem using the Sum of Interior Angles Formula or the formula for one interior angle of a polygon.  Because every polygon can be broken into triangles, you can use the Sum of the Interior Angles of a polygon formula to solve for the number of sides of your polygon.  This formula is (n-2)180 degrees.

 

First, assume the polygon has n sides, then it would also have n interior angles.  You know that each of the interior angles is 165 degrees so that gives you half of your equation, 165 times n, or 165n.

 

Second, set your interior angles equal to the interior angle formula (n-2)180 = 165n.

 

Last, solve for n.  

Use the distributive property to expand the left side of the equation:  180n - 360 = 165n

Subtract 165n from both sides of the equation and add 360 to both sides (I prefer this step to dividing by a negative number).  15n = 360

Divide by 15 to get n = 24 sides to the polygon.

 

Alternatively, you could solve an equation using the interior angle formula of 180 - (360/n) = 165.  Check your answer of 24 by substituting into this formula (n=24) to see that we got the correct answer.

 

I am available for tutoring lessons if you have more questions.

 

Answer 4

The sum of the interior angles of a regular polygon is

 

S = 180(n - 2)

 

where:

S = sum of interior angles

n = number of sides

 

In this case,

 

S = 165n

 

Plugging in this value of S into the formula,

 

165n = 180(n - 2)

 

 

Solve for n from this equation to get the number is sides.