Unit 7

Polygons

Lesson 7.1

Interior & ExteriorAngle Sumsof Polygons

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.

Definition of a Polygon• A polygon is plane figure (two-dimensional)

that meets the following conditions.1. It is formed by three or more segments called sides.2. The sides must be straight lines.3. Each side intersects exactly two other sides, one at each endpoint.4. The polygon is closed in all the way around with no gaps.5. Each side must end when the next side begins. No tails.

Polygons Not Polygons

No CurvesToo Many

Intersections

No Gaps

No Tails

Types of PolygonsNumber of Sides Type of Polygon

3

4

5

6

7

8

9

10

12

n

Triangle

QuadrilateralPentagonHexagonHeptagon

OctagonNonagonDecagon

Dodecagonn-gon

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.

Example 7.1Determine 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.

1.

No!

Yes

Concave

YesConvex

2.

Hexagon 3.

Pentagon

5.

6.

4.

Yes

ConcaveOctagon

No!

YesConcave

Heptagon

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.

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.

360o

180o

540o

720o

Theorem 11.1:Polygon Interior Angles Theorem

• The sum of the measure of the interior angles of a convex n-gon is

o180 ( 2)n

n = number of sides

Example 7.2Find the sum of the interior angles of

the following convex polygons.1.

2.

3. nonagon

4. 17-gon

Example 7.3Find x.1.

2.

3.

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

34

5

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

34

5

1 + 2 + 3 + 4 + 5 = 360o

Example 7.4Find the sum of the exterior angles of

the following convex polygons.1. Triangle

1. 3600

2. Quadrilateral2. 3600

3. Pentagon3. 3600

4. Hexagon4. 3600

5. Heptagon5. 3600

6. Dodecagon6. 3600

7. 17-gon7. 3600

Example 7.5Find x.1.

2.

3.

Lesson 7.1 Homework• Lesson 7.1 – Interior &

Exterior Angle Sums of Polygons

• Due Tomorrow

Lesson 7.2

Each Interior & ExteriorAngle of aRegular Polygon

Lesson 7.2 Objectives• Calculate the measure of each

interior angle of a regular polygon. (G1.5.2)

• 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.

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.

Example 7.6Classify the following polygons as equilateral,

equiangular, regular, or neither.1.

2.

3.

4.

5.

6.

Corollary to Theorem 11.1• The measure of each interior

angle of a regular n-gon is found using the following:

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.

o180 ( 2)n

n

Sum of the Interior Angles

Divided equally into n

angles.

Example 7.7Find the measure of each interior angle in the

regular polygons.1. pentagon

2. decagon

3. 17-gon

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:( ) ( ) ( )Angle Measure Number of Sides Sum of the Interior Angles

o( ) ( ) 180 ( 2)Angle Measure n n

Example 7.8Determine the number of sides of the

regular polygon given one interior angle.

1. 120o

2. 140o

3. 147.27o

Corollary to Theorem 11.2• Review: What is the

sum of the exterior angles of a pentagon?

• 3600

• hepatagon?• 3600

• dodecagon?• 3600

• any polygon?• 3600

• 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 1200. How many exterior angles would it take to get to the total for the exterior angles?

» 360120

» 3

• So n = 3

o360 Each Exterior Angle Measure

n

o360

n

Exterior Angle Measure

Example 7.9Find the measure of each exterior angle

of the regular polygon.1. octagon

2. dodecagon

3. 15-gon

Example 7.10Determine the number of sides of the

regular polygon given the measure of an exterior angle.

1. 72o

2. 45o

3. 27.69o

Lesson 7.2 Homework• Lesson 7.2 - Each Interior & Exterior

Angle of a Regular Polygon• Due Tomorrow

Lesson 7.3

Day 1:Area and Perimeter of Regular Polygons

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.

Theorem 11.3:Area of an Equilateral Triangle

• Area of an equilateral triangle is– A = ¼ (√3) s2

• Take ¼ times the length of a side squared and write in front of √3.– Be sure to simplify if

possible.

s

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

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 by– 360/n

• 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.

600

Apothem• The apothem is the distance

from the center to any side of the polygon.– Not to the vertex, but to the

center of the side.– The height of a triangle formed

between the center and two consecutive vertices of the polygon.

a

Theorem 11.4:Area of a Regular Polygon

• The area of a regular n-gon with side length s is half the product of the apothem and the perimeter.

• A = 1/2aP– A stands for area– a stands for apothem– P stands for perimeter of the n-

gon• Found by finding the side length and

multiplying by the number of sides

– A = 1/2a(ns)• n stands for the number of sides• s stands for the length of one side

Lesson 7.3

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

Lesson 7.3

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