The shapes below are examples of regular polygons.

Look at the sides and angles of each shape. Octagon

rectangle hexagontriangle

The following shapes are not regular polygons. Look at the sides and angles of

each of these shapes.

octagonrectangle

triangle hexagon

Can you make a conjecture (educated guess) as to what makes polygons “regular”?

regular not regular

A regular polygon is a polygon in which all sides are equal to each other, and all

angles are equal to each other.

All the sides of this octagon are the

same length and all of the angles are the

same size.

This is not a regular octagon because although its angles are all the same size, the top

and bottom of the octagon are much longer than the other

sides.

Interior Angles

The interior angles of a polygon are simply the angles inside of the figure.

A

B C

m BAC = 61°

m ABC = 59°

m BCA = 60°

We can use a protractor to find each interior angle measure of this triangle.

Notice that when we add the angles together we get 180°. This is always going to be true when you add the interior angles of a triangle together.

Since the sum of the interior angles of a triangle always equals 180°, how large is

each angle in an equilateral triangle?

180 divided by 3 = 60°

What is the sum of the interior angles of this quadrilateral?

Notice that it is two triangles put together like

this:

**Remember that the total sum of the interior angles of a triangle is always 180.

Well, you know that the sum of the interior angles of each of the triangles equals 180. Since we have two triangles we can just add 180+180 to get the total

sum of the interior angles. In this case the sum of the interior angles is 360.

Using this method we can divide any polygon into triangles by drawing in its diagonals. In order to draw diagonals go to ONE vertex of a polygon and draw all the segments possible to the other vertices.

 

Notice how the hexagon is now divided into four triangles. Now we can find the sum of the interior angles of this hexagon.

4 180 720↔ = o

 Using this method, can you think of way to find each interior angle measure for a

regular polygon?

If you have the total sum of the interior angles and you know that you have a regular polygon, you can simply divide the sum by the number of angles. 

1. Draw in the diagonals in the regular polygon.

2. Count the number of triangles formed. ( 6 in this case )

3. Multiply that number by the sum of the interior angles of a triangle.

( )4. Since all of the angles are equal,

divide that total by the number of angles. ( )

 

6 180 1080↔ = o

1080 8 135⎫ = o

Exterior Angles

Extending one side of the polygon forms the exterior angle of a polygon.

In this polygon, the exterior angle is formed by extending one side.

If we have a regular polygon, finding this exterior angle measure is a breeze. Since all of the angles are congruent, we can just divide the total by the

number of angles.

For example:

Since we are working with a regular polygon, we know that all of the exterior angles are equal.

**Remember that the exterior angle sum is always 360.

So, we can just divide 360 by 5 (since we have 5 angles) to get the measure of each exterior angle.

360 5 72⎫ = o

How would you find the exterior angle measure if you did not have a regular

polygon?

If you look at an interior and exterior angle together, you will notice that they always form a straight line.

The circled portion of the diagram shows that an interior angle plus an exterior angle form a straight line (180).

If the interior angle is 40 then we know that the exterior angle equals180-40=140