Area of Polygons and Circles

Chapter 11

11.1 Angle Measures in Polygons The sum of the measures of the interior

angles of a polygon depends on the number of sides.

Determining how many triangles are in each polygon will help you figure out the sum of the measures of the interior angles.

Sum of interior angles Draw all the diagonals from one vertex.

This will divide the polygon into triangles.

Polygon Interior Angle Theorem The sum of the measures of the interior

angles of a convex n-gon is (n-2)*180 .

The measure of each interior angle of a regular n-gon is

nn 180)2(

Examples Find the value of x.

Polygon Exterior Angle Theorem The sum of the measures of the exterior

angles of a convex polygon, one angle at each vertex, is 360 .

The measure of each exterior angle of a regular n-gon is

n360

Example Find the value of x.

Example Find the value of x.

11.2 Areas of Regular Polygons Regular Polygon: all sides are same length You know that the area of a triangle is

equal to A = ½ bh. If you are dealing with an equilateral

triangle there is a special formula: A =

(s = side)

341 2s

Example Find the area of an equilateral triangle

with 8-inch sides.

Vocabulary When dealing with a polygon, think of it as

if it were inscribed in a circle:

Vocabulary Center of a polygon: the same as the

center of the circumscribed circle Radius of the polygon: the same as the

radius of the circumscribed circle G is the center of the polygon GA is the radius

GE

D

F

C

B

A

Vocabulary Apothem of the polygon: the distance from

the center to any side of the polygon. The apothem is the segment GH.

GE

D

F

C

B

A

H

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 = aP21

6

4

Central angle of a regular polygon 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 (n) to find the measure of each central angle.

Examples Find the area of the regular octagon.

P = __________ Apothem = ___________ Area = _____

8.3

4.3

11.3 Similar Figures If two polygons are similar with the lengths

of corresponding sides in the ratio of a:b, then the ratio of their areas is a2:b2

Similar Figures The ratio of the lengths of corresponding

sides is 1:2. The ratio of the perimeters is also 1:2. The ratio of the areas is 1:4.

11.4 Circumference and Arc Length Circumference of a circle: the distance around the

circle.

Arc length: a portion of the circumference of a circle. Measure of an arc – degrees Length of an arc – linear units

The circumference C of a circle is: . .

d is the diameter of the circle r is the radius of the circle

dC rC 2

Arc Length Corollary The ratio of the length of a given arc to the

circumference is equal to the ratio of the measure of the arc to 360 .

Arc length of AB =

Arc length of AB =

360mAB

P

B

Ar2

rmAB 2360

Arc Lengths The length of a semicircle = ½ of the

circumference.

The length of a 90 arc = ¼ of the circumference.

Examples Find the length of the arc.

Examples Find the length of the arc.

Example Find the circumference.

Examples Find the measure of XY.

11.5 Areas of Circles and Sectors Area of a Circle =

Find the area of the circle.

2r

8 in.

Examples Find the diameter of the circle if the area

is 96 cm2.

Z

Sector of a Circle Sector of a Circle: the region bounded by

two radii of the circle and their intercepted arc.

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 360 .

.

3602

mABrA

Examples Find the area of the sector.

Examples A and B are two points on the circle with

radius 9 inches and m APB = 60 . Find the areas of each sector.

Finding areas of Regions

= -Area of shaded region

Area of circle

Area of polygon

Examples Find the area of the shaded region.

11.6 Geometric Probability Probability is a number from 0 to 1 that

represents the chance that an event will occur.

Geometric Probability is a probability that involves a geometric measure such as length or area.

Probability and Length Let AB be a segment that contains the

segment CD. If a point K on AB is chosen at random, then the probability that it is on CD is:

P(Point K is on CD) = Length of CD Length of AB

A C D 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:

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

J

M

Examples Find the probability that a point chosen at

random on RS is on TU. Find the probability that a point chosen at

random on RS is on TU.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

R T U S

Examples Find the probability that a randomly

chosen point in the figure lies in the shaded region.

Examples Find the probability that a randomly

chosen point in the figure lies in the shaded region.