Download Chapter 11 Notes

Chapter 11 Notes
Mrs. Myers – Geometry
Name ______________________________
Period ______
11.1 Angle Measures in Polygons
Polygon
# of Sides
Number of
Triangles
Sum of Measures
of Interior Angles
Triangle
Quadrilateral
Pentagon
Hexagon
…
N-gon
* Theorem 11.1: Polygon Interior Angles = the sum of the measures of the interior
angles of a convex n-gon is
 n  2 180
* Corollary to Theorem 11.1: the measure of each interior angle of a regular n-gon is
 n  2
180
n
* Theorem 11.2: Polygon Exterior Angles = the sum of the measures of the exterior
angles of a convex polygon, one angle at each vertex is 360 .
* Corollary to Theorem 11.2: the measure of each interior angle of a regular n-gon is
360
n
Ex. 1 Find x.
114
105
102
x
135
Ex. 2 The measure of each interior angle of a regular polygon is 165 . How many sides
does the polygon have?
Ex. 3 Find y.
2y 
y
y
2y 
Ex. 4 Find x.
11.2 Areas of Regular Polygons
* Theorem 11.3: Area of an Equilateral Triangle =
A
1 2
s 3
4

Center of Polygon: center of its circumscribed circle.

Radius of the Polygon: radius of its circumscribed circle.

Apothem of the Polygon: the distance from the center to any side of the polygon
(it is also the height of a triangle between the center and 2 consecutive vertices of
the polygon…so it must hit at a right angle).
* Theorem 11.4: Area of a Regular Polygon =
A
1
ap
2
a  apothem
p  perimeter of the polygon

Central Angle of a Regular Polygon: is an angle whose vertex is the center and
whose sides contain 2 consecutive vertices of the polygon.

Measure of Each Central Angle:

360
n
Ex. 1 Find the area of the triangle.
Ex. 2 Find the area of an equilateral triangle whose perimeter is 6 in.
Ex. 3 The bottom of a glass is a regular 12-gon with a side length of 1.2 cm and a
radius of 2.3 cm. What is the area of the bottom of the glass?
Ex. 4 A regular octagon is inscribed in a circle with a radius of 4 units. Find the area of
the octagon.
11.3 Perimeters and Areas of Similar Figures
* Theorem 11.5: Areas of Similar Polygons = if 2 polygons are similar with the lengths
of corresponding sides in the ratio a : b , then the ratio of their areas is a 2 :b 2 .
ka
kb
I
II
If I II , then
side length of I
ka
a


and
side length of II
kb
b
Perimeter of I
a

and
Perimeter of II b
area of I
a2
 2
area of II
b
Ex. 1 The ratio of corresponding sides of 2 similar hexagons is 8:2 .
A) Find the ratio of their perimeters
B) Find the ratio of their areas.
Ex. 2 A rectangular tablecloth is 60 in by 120 in. A rectangular place mat made from
the same cloth is 12 in. by 24 in. and cost $5.00. Compare the areas of the place mat and
table cloth to find a reasonable cost for the tablecloth.
Ex. 3 A store sells trampolines that are all regular hexagons. One trampoline has a side
length of 4.5 ft and an area of 52.6 ft 2 . Find the area of the trampoline whose perimeter
is 67.5 ft.
11.4 Circumference and Arc Length

Circumference: of a circle is the distance around the circle.
* Theorem 11.6: Circumference of a Circle =
C  d or C  2r
Ex. 1 Find the circumference of a circle with a radius of 9 in.
Ex. 2 Find the radius of a circle with a circumference of 52 in.

Arc Length: is a portion of the circumference of a circle.
* Arc Length Corollary:
arc length of AB 
 m AB   2r 
360
Ex. 2 Find the length of each arc.
A)
B)
16 in
M
60
7 cm
100
M
K
B
Ex. 3 Find the indicated measure.
A) mLM
B) Circumference
8 in.
L
45
M
16. 76 in
10.2 cm
11.5 Areas of Circles and Sectors
* Theorem 11.7: Area of a Circle =
A  r 2
Ex. 1
A) Find the area of a circle with a radius of 3 m.
B) Find the diameter of circle B when the area is 254.5 ft 2 .

Sector of a Circle: is the region bounded by two radii of the circle and their
intercepted arc.
* Theorem 11.8: Area of a Sector =
 m AB   r  
A
2
360
Ex. 2 Find the area of the sector.
Ex. 3 L and M are two points on circle R with radius of 50 cm and m LRM  150 .
Find the area of the sectors formed by
LRM .
Ex. 4 Find the area of the shaded region.
A)
B)
11.6 Geometric Probability

Probability: a number from 0 to 1 that represents the chance that an event will
occur.

Geometric Probability:
o Probability and Length:
A
C
D
B


P Po int K is on CD 
length of CD
length of AB
o Probability and Area:
M
J
P  po int K is in region M  
area of M
area of J
Ex. 1 On the number line AF : The probability of a point landing in:
 
A) P AB
 
B) P BD
 
C) P BF
Ex. 2 Find the P  hitting the bull ' s eye 
17 in
3 in
17 in
Ex. 3 Find the
M
13 m
9 m
3 m
B
14 m
B) P  placing a po int in M 
C) P  Placing a po int not in M 