**Chapter 11.1 Angle Measures in Polygons:**

**I) lets look at the sum of polygon angle measures:**

**Polygon - Number of sides - Number of triangles can make from the polygon - sum of the measures of the interior angles:**

A) Triangle - 3 sides - 1 triangle - 180°

B) Quadrilateral - 4 sides - 2 triangles - 360°

C) Pentagon - 5 sides - 3 triangles - 540°

D) Hexagon - 6 sides - 4 triangles - 720°

E) Heptagon - 7 sides - 5 triangles - 900°

F) Octagon - 8 sides - 6 triangles - 1080°

G) Nonagon - 9 sides - 7 triangles - 1260°

H) Decagon - 10 sides - 8 triangles - 1440°

I) Undecagon - 11 sides - 9 triangles - 1620°

J) Dodecagon - 12 sides - 10 triangles - 1800°

Do you see a pattern? The number of sides minus 2 is equal to the number of triangles so...

**II) Theorem:**

**A) Polygon Interior Angles Theorem**- the sum of the measures of the interior angles of a convex n-gon is (n - 2) (180°)

**Example:**What is the sum of the measures of the interior angles of a convex 20-gon?

(20 - 2)(180°) = 3240°

**B)**The measures of each interior angle of a regular n-gon is:

(1/n)(n-2)(180°)

**Example:**Given a regular hexagon, what is the measure of each angle?

(1/6)(6-2)(180°) = (1/6)(4)(180°) = 120°

**C) Polygon Exterior Angles Theorem**- the sum of the measures of the exterior angles of a convex polygon, one angle iat each vertex, is 360°.

**Example:**If the exterior angles of a polygon is x, 2x, 2x, 4x, and 3x, solve for x.

x + 2x + 2x + 4x + 3x = 360

12 x = 360

x = 30

**Example:**Given a regular polygon has 20 sides, what is the measure of an exterior angle and an interior angle?

360/20 = 18, so every exterior angle is 18°

180° - 18° = 162° , so every interior angle is 162°

(20 - 2)(180) = (162)(20)

3240 = 3240 so it checks

**Example:**Given the measrue of each interior angle of a regular n-gon is 120°, how many sides does the polygon have?

(1/n)(n-2)(180) = 120

(n-2)(180) = 120n

180n - 360 = 120n

60n = 360

n = 6

If the interior angle is 120°, then the exterior angle is 60°

360/60 = 6 sides, same answer!

**Chapter 11.2 Areas of Regular Polygons:**

**I) Area of an Equilateral Triangles is:**

A = ((√3)/4) s

^{2}

Example: Given an equilateral triangle with side length of 4 what is the area?

A = ((√3)/4)(4

^{2})

A = ((√3)/4)(16)

A = 4√3 square units

**II) Vocabulary:**

**A) Center of the polygon**- when you circumscribe a circle about a regular polygon, the center point of the circle is the center of the polygon.

**B) Radius of the polygon**- when you circumscribe a circle about a regular polygon, from the center point of the circle to the vertex of the polygon is the radius of the polygon.

**C) Apothem of the polygon**- the distance from the center point of the polygon perpendicular to any side of the polygon.

**D) A Central angle of a regular polygon**- is an angle whose vertex is the center and whose sides contain two consecutive vertices of the polygon. You can divide 360° by the number of sides to find the measure of each central angle of the polygon.

**III) Theorem:**

**A) Area of a Regular Polygon**- the area of a regular n-gon with side length "s" is half the product of the apothem "a" and the perimeter "P"

A = (1/2)(a)(P)

**Chapter 11.3 Perimeters and Areas of Similar Figures**

I) Theorems:

I) Theorems:

**A) Areas of Similar Polygons**- if two polygons are similar with the lengths of the corresponding sides in the ratio of a:b, then the ratio of their areas is a

^{2}:b

^{2}

**Example:**Given two similar hexagons with corresponding sides of 4 and 5, what is the ratio of their areas?

4

^{2}:5

^{2}

16:25

**Chapter 11.4 Circumference and Arc Length**

A) Circumference- of a circle is the distance around the circle.

A) Circumference

C = 2π r = πd

**B) Arc length corollary:**

in a circle, the ratio of the length of a given arc to the circumference is equal to the ratio of the measure of the arc to 360º

**Example:**recall, the central angle is equal to it's intercepted arc

Given circle P with central angle APB = 30º and radius of 8 radius, what is the measure of the intercepted arc AB?

(measure of the arc AB)/(2π8) = (30/360)

(measure of the arc AB)/ (16π) = (1/6)

measure of the arc AB = 16π/6

measure of the arc AB = 8π/3 inches or about 8.37758041 inches

**Chapter 11.5 Areas of Circles and Sectors**

**I) Vocabulary:**

**A) Sector of a Circle**- is the region bounded by two radii of the circle and their intercepted arec.

**II) Theorems:**

**A) Area of a Circle**- the are of a circle is π times the square of the radius

A = πr

^{2}

**Example:**given a circle with radius of 9 cm, what is the area?

A = π(9)

^{2}= 81π square cm.

**B) Area of a Sector**- the ratio of the area A of a sector of a circle to the area of the circle is equal to the ratio of the measure of the intercepted arc to 360º

Example: Given Circle C with points A and B on the circle, the major arc AB is 293º and the radius is 10 cm, what is the area of the sector?

(Area of sector)/(π 10

^{2}) = 293º /360º

Area of sector/100π = 293/360

Area of sector = 1465π/18 which is about 255.6907354 square cm.

**Chapter 11.6 Geometric Probability**

**I) Vocabulary**

**A) Probability**is a number from 0 to 1 that represents the chance that an event will occur.

**B) Geometric Probability**- involves geometric measures such as length or area

**II) Theorems:**

**A) Probability and length**- Let line segment AB be a segment that contains the segment CD. If a point K on line segment AB is chosen at random, then the probability that it is on CD is as follows:

P(Point K is on line segment CD) = (length of CD)/(length of AB)

**Example:**Given AB = 5 and CD = 2, then the probability that point K is on CD is

5/2

**B) Probability and Area**- Let J be a region that contains region M. If a point K in J is chosen at random, then the probability that it is in region M is as follows:

P(Point K is in region M) = (Area of M)/(Area of J)

**Example:**You mow a rectangular lawn of 20 feet by 30 feet and it has a garden of 5 feet by 6 feet. What is the geometric probability that if someone tosses a ball that it will land in the garden?

(5)(6) = 30 square feet

(20)(30) = 600 square feet so

30/600 = 1/20