1. Geometry Day 41
Polygons

2. Agenda Polygons Regular Polygons Definition Interior angles
Exterior angles
Regular Polygons

3. Polygons
A polygon is a two-dimensional closed plane figure composed of line segments.
There are no curves in a polygon.
Polygons are named after the number of sides they have.

4. Common Polygons Some common polygons are:
Triangle – 3 sides
Quadrilateral – 4 sides
Pentagon – 5 sides
Hexagon – 6 sides
Heptagon – 7 sides
Octagon – 8 sides
Nonagon – 9 sides
Decagon – 10 sides
Other polygons have names, but they can be hard to remember. Most people abbreviate them as 11-gon, 12-gon, 35-gon, etc.

5. Convexity
A polygon is convex if each of its interior angles is less than 180.
This is a convex decagon.
If one or more interiors angles is greater than 180, then the polygon is non-convex, or concave.
This is a concave octagon.

6. Interior Angles
A diagonal in a polygon connects two non-adjacent vertices.
Complete the first handout.
Starting from the dot in each figure, draw as many diagonals as you can.
Count the number of triangles that you created, and use them to determine the sum of all the interior angles of each figure.

7. Polygon Interior Angle Sum Theorem
The sum of the interior angles of any polygon is:
(n – 2)180
where n = number of sides
How many degrees in a 20-gon?
What is the sum of the interior angles of a 52-gon?
A polygon has 1980. How many sides does it have?
A polygon has 2455. How many sides does it have?

8. Exterior Angles
An exterior angle of a polygon is formed by extending a straight line from one of the interior angles.
An exterior angle will always be supplementary to its adjacent interior angle.
Complete the second handout. Look for patterns.

9. Polygon Exterior Angle Sum Theorem
The sum of the exterior angles of any polygon is:
360
Proof…
What is the sum of the exterior angles of a 38-gon?

10. Diagonals
Let’s go back to page 1. This time draw all of the diagonals possible for each shape. Make a table with the total number of diagonals for each polygon.
See if you can generate a formula that expresses this.
The number of unique diagonals of a polygon is:
n(n – 3) 2

11. Regular Polygons
An equilateral polygon is one where all its sides are congruent.
An equiangular polygon is one where all its angles are congruent.
A regular polygon is one where all its sides are congruent and all its angles are congruent.

12. Angles of Regular Polygons
Given what we know about the interior and exterior angles of polygons in general, and given that all the angles of a regular polygon are the same, can you generate formulas for the individual interior and exterior angles of a regular polygon?

13. Angles of Regular Polygons
Each interior angle of a regular polygon is:
(n – 2)180 n
Each exterior angle of a regular polygon is:
360

14. Angles of Regular Polygons
What is the measure of each interior angle of a regular decagon?
What is the measure of each exterior angle of a regular 20-gon?
A regular polygon has an interior angle of 120. How many sides does it have?
A regular polygon has an exterior angle of 10. How many sides does it have?

15. Homework 24
Workbook, p. 72
Handout