1. Unit 7
Polygons

2. Interior & Exterior Angle Sums of Polygons
Lesson 7.1
Interior & Exterior
Angle Sums
of Polygons

3. Lesson 7.1 Objectives
Calculate the sum of the interior angles of a polygon. (G1.5.2)
Calculate the sum of the exterior angles of a polygon. (G1.5.2)
Classify different types of polygons.

4. Definition of a Polygon
A polygon is plane figure (two-dimensional) that meets the following conditions.
It is formed by three or more segments called sides.
The sides must be straight lines.
Each side intersects exactly two other sides, one at each endpoint.
The polygon is closed in all the way around with no gaps.
Each side must end when the next side begins. No tails.
Polygons
Not Polygons
Too Many
Intersections
No Curves
No Gaps
No Tails

5. Types of Polygons Triangle Quadrilateral Pentagon Hexagon Heptagon
Number of Sides
Type of Polygon 3 4 5 6 7 8 9 10 12 n
Triangle
Quadrilateral
Pentagon
Hexagon
Heptagon
Octagon
Nonagon
Decagon
Dodecagon
n-gon

6. Concave v Convex
A polygon is convex if no line that contains a side of the polygon contains a point in the interior of the polygon.
Take any two points in the interior of the polygon. If you can draw a line between the two points that never leave the interior of the polygon, then it is convex.
A polygon is concave if a line that contains a side of the polygon contains a point in the interior of the polygon.
Take any two points in the interior of the polygon. If you can draw a line between the two points that does leave the interior of the polygon, then it is concave.
Concave polygons have dents in the sides, or you could say it caves in.

7. Example 7.1 Yes No! Concave Octagon Yes No! Concave Hexagon Concave
Determine if the following are polygons or not. If it is a polygon, classify it as concave or convex and name it based on the number of sides.
No!
Yes
Concave
Octagon
Yes
No!
Concave
Hexagon
Concave
Yes
Heptagon
Yes
Convex
Pentagon

8. Diagonals of a Polygon
A diagonal of a polygon is a segment that joins two nonconsecutive vertices.
A diagonal does not go to the point next to it.
That would make it a side!
Diagonals cut across the polygon to all points on the other side.
There is typically more than one diagonal.

9. Interior Angles of a Polygon
The sum of the interior angles of a triangle is
180o
The sum of the interior angles of a quadrilateral is
360o
The sum of the interior angles of a pentagon is
???
The sum of the interior angles of a hexagon is
By splitting the interior into triangles, it should be able to tell you the sum of the interior angles.
Pick one vertex and draw all possible diagonals from that vertex.
Then, count up the number of triangles and multiply by 180o.
180o
360o
540o
720o

10. Theorem 11.1: Polygon Interior Angles Theorem
The sum of the measure of the interior angles of a convex n-gon is
n = number of sides

11. Example 7.2
Find the sum of the interior angles of the following convex polygons.
nonagon
17-gon

12. Example 7.3
Find x.

13. Exterior Angles
An exterior angle is formed by extending each side of a polygon in one direction.
Make sure they all extend either pointing clockwise or counter-clockwise. 1 2 3 4 5

14. Theorem 11.2: Polygon Exterior Angles Theorem
The sum of the measures of the exterior angles of a convex polygon is 360o.
As if you were traveling in a circle! 1 2 3 4 5
 1 +  2 +  3 +  4 +  5 = 360o

15. Example 7.4
Find the sum of the exterior angles of the following convex polygons.
Triangle
3600
Quadrilateral
Pentagon
Hexagon
Heptagon
Dodecagon
17-gon

16. Example 7.5
Find x.

17. Lesson 7.1 Homework
Lesson 7.1 – Interior & Exterior Angle Sums of Polygons
Due Tomorrow

18. Each Interior & Exterior Angle of a Regular Polygon
Lesson 7.2
Each Interior & Exterior
Angle of a
Regular Polygon

19. Lesson 7.2 Objectives
Calculate the measure of each interior angle of a regular polygon. (G1.5.2)
Determine the number of sides of a regular polygon based on the measure of one interior angle.
Determine the number of sides of a regular polygon based on the measure of one exterior angle.

20. Regular Polygons
A polygon is equilateral if all of its sides are congruent.
A polygon is equiangular if all of its interior angles are congruent.
A polygon is regular if it is both equilateral and equiangular.
Remember: EVERY side must be marked with the same congruence marks and EVERY angle must be marked with the same congruence arcs.

21. Example 7.6
Classify the following polygons as equilateral, equiangular, regular, or neither.

22. Corollary to Theorem 11.1
The measure of each interior angle of a regular n-gon is found using the following:
Sum of the Interior Angles
Divided equally into n angles.
It basically says to take the sum of the interior angles and divide by the number of sides to figure out how big each angle is.

23. Example 7.7
Find the measure of each interior angle in the regular polygons.
pentagon
decagon
17-gon

24. Finding the Number of Sides
By knowing the measurement of one interior angle of a regular polygon, we can determine the number of sides of the polygon as well.
How?
Since we know that all angles are going to have the same measure we will multiply the known angle by the number of sides of the polygon.
That will tell us how many sides it would take to be set equal to the sum of all the interior angles of the polygon.
However, since we do not know the number of sides of the polygon, nor do we know the total sum of the interior angles of that polygon we are left with the following formula to work with:

25. Example 7.8
Determine the number of sides of the regular polygon given one interior angle.
120o
140o
147.27o

26. Corollary to Theorem 11.2
Review: What is the sum of the exterior angles of a pentagon?
3600
hepatagon?
dodecagon?
any polygon?
Then how would we find the measure of an exterior angle if it were a regular polygon?
Divide 360o by the number of exterior angles formed.
Which happens to be the same as the number of sides (n).
This can also be worked in “reverse” to determine the number of sides of a regular polygon given the measure of an exterior angle.
How?
Figure out how many times that angle measure would go into 360o.
Say each exterior angle is How many exterior angles would it take to get to the total for the exterior angles?
360120 3
So n = 3 

27. Example 7.9
Find the measure of each exterior angle of the regular polygon.
octagon
dodecagon
15-gon

28. Example 7.10
Determine the number of sides of the regular polygon given the measure of an exterior angle.
72o
36o
27.69o

29. Lesson 7.2 Homework
Lesson Each Interior & Exterior Angle of a Regular Polygon
Due Tomorrow

30. Day 1: Area and Perimeter of Regular Polygons
Lesson 7.3
Day 1:
Area and Perimeter of Regular Polygons

31. Lesson 7.3 Objectives
Calculate the measure of the central angle of a regular polygon.
Identify an apothem
Calculate the perimeter and area of a regular polygon. (G1.5.1)
Utilize trigonometry to find missing measurements in a regular polygon.

32. Parts of a Polygon
The center of a polygon is the center of the polygon’s circumscribed circle.
A circumscribed circle is one in that is drawn to go through all the vertices of a polygon.
The radius of a polygon is the radius of its circumscribed circle.
Will go from the center to a vertex. r

33. Central Angle of a Polygon
The central angle of a polygon is the angle formed by drawing lines from the center to two consecutive vertices.
This is found in a regular polygon by:
That is because the total degrees traveled around the center would be like a circle.
Then divide that by the number of sides because that determines how many central angles could be formed.

34. Example 7.11 Find the central angle of the following regular polygons.
pentagon
heptagon
decagon
18-gon

35. Reminder of - Postulate 22: Area of a Square Postulate
The area of a square is the square of the length of its side. s

36. Theorem 11.3: Area of an Equilateral Triangle
Area of an equilateral triangle is: s

37. Example 7.12
Find the area of the equilateral triangles.

38. Interior Triangles of a Hexagon
A regular hexagon is unique in that it is the only polygon whose central angles form vertices of interior triangles that are all equilateral.
Remember, an equilateral triangle is also regular!
Remember that every side of the interior triangles form a radius of the circle around the vertices. 4 5 3
AND ALL RADII ARE CONGRUENT! 6 1 2
So to find the area of a regular hexagon would be like finding the area of SIX equilateral triangles!

39. Example 7.13
Find the area of the regular hexagons.

40. Lesson 7.3a Homework
Lesson 7.3 – Area & Perimeter of Regular Polygons (Day 1)
Due Tomorrow 40

41. Lesson 7.3 Day 2: Area and Perimeter of Regular Polygons
(Using Special Triangles & Trigonometry)

42. Perimeter of a Regular Polygon
Recall that the perimeter is the sum of the lengths of all the sides of a figure.
Well what is true about the side lengths of a regular polygon?
They are all equilateral.
So the quickest and best way to find the perimeter when all sides are congruent is:
s = side length
n = number of sides 42

43. Do the Equilateral Triangles Still Exist?
What is true about equilateral triangles?
All sides are congruent, and…
All angles are congruent.
And each angle must be 60o
What is the central angle of a regular pentagon?
72o
Would that central angle help to form an equilateral triangle?
No, because all angles must be 60o.
What is the central angle of a regular heptagon?
51.43o
So the equilateral triangles are only formed in hexagons.
Therefore, there must be another way to find the area of other regular polygons.

44. Apothem
The apothem is the length of a line segment in a regular polygon drawn:
From the center of the polygon to one of its sides.
Such that it is perpendicular to the side.
And it bisects the side of the polygon. a

45. Theorem 11.4: Area of a Regular Polygon
The area of a regular polygon is found using:
P = perimeter
P = n•s
And how do we find the perimeter?
a = apothem
( )
s = side length
n = number of sides

46. Example 7.14
Find the perimeter and area of the regular polygons.

47. Lesson 7.3b Homework
Lesson 7.3 – Area & Perimeter of Regular Polygons (Day 2)
Due Tomorrow 47

48. Lesson 7.3 Day 3: Area and Perimeter of Regular Polygons
(Using Special Triangles & Trigonometry - Again)

49. The Apothem and the Central Angle
Remember it is necessary to know the length of the apothem when finding the area of a regular polygon.
A = 1/2a•n•s
So what would happen if the length of the apothem was unknown?
Hint: Draw the central angle and what do you see?
Because the apothem is a perpendicular bisector to the side of known length
It divides the side in half, and
It divides the central angle in half. a 49

50. Finding the Area with Only a Known Side Length
To find the area of a regular polygon with only a known side length, you must also know the length of the apothem.
To do so, create a small right triangle using:
The apothem.
Half of the central angle.
Half of the given side length.
And then use trigonometry to solve for the unknown apothem.
SOH CAH TOA a 6
Half the given side length 50

51. Example 7.15
Find the area of the regular polygons. 51

52. Finding the Area with Only a Known Apothem
To find the area of a regular polygon with only a known apothem, you must also know the side length.
To do so, create a small right triangle using:
The apothem.
Half of the central angle.
Half of the given side length.
And then use trigonometry to solve for the unknown side length.
SOH CAH TOA
Don’t forget that you just found half the side length. So DOUBLE it!
4.5 x 52

53. Example 7.16
Find the area of the regular polygons. 53

54. Lesson 7.3b Homework
Lesson 7.3 – Area & Perimeter of Regular Polygons (Day 3)
Due Tomorrow 54