Download Chapter 1: Shapes and Transformations

In previous chapters, you have extensively studied triangles and quadrilaterals to learn more about their sides
and angles. In this chapter, you will broaden your focus to include polygons with 5, 8, 10, and even 100 sides.
You will develop a way to find the area and perimeter of a regular polygon and will study how the area and
perimeter changes as the number of sides increase.
In the second half of the chapter, you will re-examine similar shapes to study what happens to the area and
perimeter of a shape when it is enlarged or reduced. Finally, at the end of the chapter you will connect your
understanding of polygons with your knowledge of the area ratios of similar figures to find the area and
circumference of circles of all sizes.
In this chapter, you will learn:
 about special types of polygons, such as regular and non-convex polygons
 how the measure of the interior and exterior angles of a regular polygon are related to the number of
sides of the polygon
 how the areas of similar figures are related
 how to find the area and circumference of a circle and parts of circles and use this ability to solve
problems in various contexts.
pg.1
8.1 – How Can I Build It?_________________________________
Pinwheels and Polygons
In previous chapters, you have studied triangles and quadrilaterals. In Chapter 8, you will broaden your focus
to include all polygons and will study what triangles can tell us about shapes with 5, 8, or even 100 sides.
By the end of this lesson, you should be able to answer these questions:
 How can you use the number of sides of a regular polygon to find the measure of the central angle?
 What type of triangle is needed to form a regular polygon?
8.1 – PINWHEELS AND POLYGONS
Itzel loves pinwheels. One day in class, she noticed that if she put three congruent triangles together, that one
set of the corresponding angles are adjacent, she could make a shape that looks like a pinwheel.
a. Can you determine any of the angles of her triangles? Explain how you found your answer.
b. The overall shape (outline) of Itzel's pinwheel is shown at right. How many sides does it
have? What is another name for this shape?
c. Itzel's shape is an example of a polygon because it is a closed, two dimensional figure made of
straight line segments connected end-to-end. As you study polygons in this course, it is useful to
use these names because they identify how many sides a particular polygon has. Some of these
words may be familiar, while others may be new. Fill in the names of the polygons below. Then,
draw an example of a heptagon.
Name of Polygon
# of sides
Name of Polygon
# of sides
3
8
4
9
5
10
6
12
7
n
8.2 – MAKING PINWHEELS
Itzel is very excited. She wants to know if you can build a pinwheel using any angle of her triangle. Obtain a
set of triangles from your teacher. Work with your team to build pinwheels and polygons by placing different
corresponding angles together at the center. You will need to use the triangles from all four team members
together to build one shape. Be ready to share your results with the class.
Angle #
# of triangles
needed
Measure of the central angle
1
2
3
pg.2
8.3 –PINWHEEL PATTERNS
Jorge likes Itzel's pinwheels but wonders, "Will all triangles build a pinwheel or a polygon?"
a. Use the different triangles provided by your teacher. Work together to determine which congruent triangles
can build a pinwheel (or polygon) when corresponding angles are placed together at the center. If it works, fill
in the table.
Angle #
# of triangles
needed
Measure of the central angle
Polygon name
A
B
C
D
E
F
b. Explain why one triangle may be able to create a pinwheel or polygon while another triangle cannot.
c. Jorge has a triangle with interior angle measures 32, 40, and 108. Will this triangle be able to form a
pinwheel? Explain? If so, at what angle?
8.4 –POLYGONS
Jasmine wants to create a pinwheel with equilateral triangles.
a. How many equilateral triangles will she need? Explain how you know.
b. What is the name for the polygon she created?
c. Jasmine's shape is an example of a convex polygon, while Inez's shape, shown at right is
non-convex (or concave). Study the examples below and write a definition of a convex
polygon on your paper.
pg.3
8.5 –CONCAVE VS. CONVEX
Brenda noticed that the non-convex (concave) shapes all had a part that went inward, like a cave. She decided
to investigate more. Sort the shapes below as either "convex" or "concave".
8.6 –CONCLUSIONS
How can you tell when a shape is convex? How can you find the central angle of a regular polygon?
pg.4
8.2 – What is Its Measure?_________________________________
Interior Angles of a Polygon
In an earlier chapter, you discovered that the sum of the interior angles of a triangle is always 180. But what
about the sum of the interior angles of other polygons, such as hexagons or decagons? Does it matter if the
polygon is convex or not? Consider these questions today as you investigate the angles of a polygon.
8.7 – INTERIOR ANGLES
With your team, find the sum of the measures of its interior angles as many ways as
you can. You may want to use the fact that the sum of the angles of a triangle is
180. Be prepared to share your method with the class.
8.8 – SUM OF THE INTERIOR ANGLES OF A POLYGON
In the previous problem, you found the sum of the measures of the interior angles of a pentagon. But what
about other polygons?
a. Use your method from the previous problem to find the sum of the interior angles of other polygons. Draw a
sketch to help support your answers. Find the sum of the measures of several shapes. (It might be helpful to
split up the tasks within your group).
b. Complete the chart below based on your findings.
# of sides
of polygon
# of
triangles
Sum of
interior
angles
3
4
5
6
7
8
9
10
15
25
50
100
1
180
c. What if it was an n-gon? What is the sum of the interior angles then?
pg.5
8.9 – FINDING ANGLES
Use the angle relationships in each of the diagrams below to solve for the given variables. Show all work.
8.10 – CENTRAL VS INTERIOR ANGLES
Below is a list of regular polygons. Find the measure of each central angle and each interior angle. Show all
work.
a
A regular 40-gon
y
b
x
Central angle: ____________
Central angle: ____________
Central angle: ____________
Interior angle: ____________
Interior angle: ____________
Interior angle: ____________
8.11 – CONCLUSIONS
How do you find the sum of the interior angles? How is this measure different than the central angle?
Sum of the Interior
Angles
Each interior angle in
a regular polygon
pg.6
8.3 – What If It Is A Regular Polygon?___________________________
Angles of a Regular Polygon
In the last section, you discovered how to determine the sum of the interior angles of a polygon with any
number of sides. But what more can you learn about a polygon? Today you will focus on the interior and
exterior angles of regular polygons.
8.12 – EXTERIOR ANGLES
Jeremy asks, "What about exterior angles? What can we learn about them?"
a. Examine the regular hexagon shown at right. Angle a is an example of an exterior
angle because it is formed on the outside of the hexagon by extending one of its sides.
Are all of the exterior angles of a regular hexagon equal? Explain how you know.
x
x
x
x
x
x
x
b. Find a. Be prepared to share how you found your answer.
c. This regular hexagon has six exterior angles, as shown in the diagram above. What is the sum of the exterior
angles of a regular hexagon?
d. What about the exterior angles of other regular polygons? Explore this with your team. Have each team
member choose a different shape from the list below to analyze. For each shape find the measure of one
exterior angle and the sum of the exterior angles.
(1) equilateral triangle
Each exterior
Sum of exterior
(2) regular decagon
Each exterior
Sum of exterior
(2) regular octagon
Each exterior
Sum of exterior
(4) regular dodecagon (12-gon)
Each exterior
Sum of exterior
e. Compare your results from part (d). As a team, complete the conjecture below.
The sum of the exterior angles of a polygon is _____________.
f. Is your conjecture from part (e) true for all polygons or for only regular polygons? Does it matter if the
polygon is convex? Explore these questions using a dynamic tool. Write a statement about your findings.
pg.7
8.13 – REGULAR, EQUILATERAL, EQUIANGULAR
Fern states, "If a triangle is equilateral, then all angles have equal measure and it must be a regular polygon."
Does this logic work for polygons with more than 3 sides?
a. If all the sides of a polygon (such as a quadrilateral) are equal, does that mean that the angles must be equal?
If you can, draw a counterexample.
b. What if all the angles are equal? Does that force a polygon to be equilateral? Explain your thinking. Draw
a counterexample if possible.
8.14 – MISSING ANGLES
Find the missing angles of the regular polygons below. Be ready to justify your answers.
c
a
y
x
v
z w
b
r
m n
p
pg.8
8.15 – USING INTERIOR AND EXTERIOR ANGLES
Use your understanding of polygons to answer the questions below, if possible. If there is no solution, explain
why not.
a. Gerardo drew a regular polygon that had exterior angles measuring 40. How many sides did his polygon
have? What is the name for this polygon?
b. A polygon has an interior angle sum of 2,520. How many sides does it have?
c. A quadrilateral has four sides. What is the measure of each of its interior angles?
d. What is the measure of an interior angle of a regular 360-gon? Is there more than one way to find this
answer?
8.16 – CONCLUSIONS
Complete the chart with the correct formulas needed to find the missing angles. How does the formula for the
exterior angles compare to the formula for the central angles?
Interior
Exterior
Sum
Each (regular)
pg.9
8.4 – What is the Connection?___________________________
Regular Polygon Angle Connections
During this chapter, you have discovered many ways the number of sides of a regular polygon is related to the
measures of the interior exterior angles of the polygon. How can these relationships be useful? And what is the
most efficient way to go from one measurement to another? This lesson will explore these questions so that you
will have a complete set of tools to analyze the angles of a regular polygon.
8.17 – CONNECTIONS
What formulas do you already have? Which do you still need? Explore this as you answer the questions below.
a. Find the measurements of an interior angle of a 15-gon.
b. What if you know that the measure of an interior angle of a regular polygon is 162? How many sides must
the polygon have?
c. If you know the number of sides of a regular polygon, how can you find the measure of an exterior angle
directly? Find the measurements of an exterior angle of a decagon.
d. If the measure of an exterior angle of a regular polygon is 15, how many sides does it have? What is the
measure of an interior angle? Show work.
8.20 – EITHER WAY
Suppose a regular polygon has an interior angle measuring 120. Find the number of sides using two different
strategies. Show all work. Which strategy was most efficient?
pg.10
8.21 – EXTRA PRACTICE
Use your knowledge of polygons to answer the questions below.
a. How many sides does a polygon have if the sum of the measures of the interior angles is 1980? 900?
b. If the exterior angle of a regular polygon is 90, how many sides does it have? What is the name of this
shape?
c. Each interior angle of a regular pentagon has measure 2x + 4. What is x?
d. The measure of four of the exterior angles of a pentagon are 57, 74, 56, and 66. What is the measure of
the remaining angle?
e. Find the sum of the interior angles of an 11-gon. Does it matter if it is regular or not?
8.22 – CONCLUSIONS
How can using the exterior angle help you
find the interior angle? How are they related?
pg.11
8.5 – What is the Area?___________________________________
Finding the Area of Regular Polygons
In this chapter you have developed a method to find the measures of the interior and exterior angles of a regular
polygon. How can this be useful? Today you will use what you know about the angles of a regular polygon to
explore how to find the area of any regular polygon with n sides.
8.23 – MULTIPLE STRATEGIES
With your team, find the area of each shape below. Make sure that your results from using different strategies
are the same. Make sure everyone on your team agrees.
8.24 – EXTRA PRACTICE
Find the area of the two regular polygons below. Look for special triangles or SOH-CAH-TOA to help find the
missing lengths.
pg.12
8.25 – WRITING THE DIRECTIONS
Susan is confused. Explain to her how to find the area of any regular polygon. Explain each step clearly so that
it will work on any shape.
8.26 – FERTILIZER
Beth needs to fertilize her flowerbed, which is in the shape of a regular pentagon. A bag of
fertilizer states that it can fertilize up to 150 square feet, but Beth is not sure how many bags
of fertilizer to buy.
Beth does know that each side of the pentagon is 16 feet long. Find the area of the flowerbed
and tell Beth how many bags of fertilizer to buy.
16ft
8.27 – CONCLUSIONS
Examine the steps to find the area of the
polygon below. How does this compare to
your directions?
1.
2.
3.
4.
5.
Find central angle (360/n)
Divide by 2 for height splitting angle in half
Find the base and/or height of triangle
Find the area of the triangle
Multiply by the number of triangles
h
pg.13
8.6 – How does the Area Change?_____________________________
Finding the Area of Regular Polygons
Much of this course has focused on similarity. In Chapter 3, you investigated how to enlarge and reduce a
shape to create a similar figure. You also have studied how to use proportional relationships to find the
measures of sides of similar figures. Today you will study how the areas of similar figures are related. That is,
as a shape is enlarged or reduced in size, how does the area change?
8.30 – MIGHTY MASCOT
To celebrate the victory of your school's championship girls' gymnastics team, the student body has decided to
hang a giant flag with your school's mascot on the gym wall. To help design the flag, your friend Archie has
created a scale version of the flag measuring 1ft wide and 2ft tall.
a. The student body would like the final flag to be 3 feet tall. How wide will the final flag be? Justify your
solution?
b. If Archie used $2 worth of cloth to create his scale model, then how much will the cloth cost for the fullsized flag? Discuss this with your team. Explain your reasoning.
c. Obtain the Resource Page from your teacher. How many small flags did it take to fit into the large flag?
Does this confirm your answer in part (b)? If not, what will the cloth cost for the flag?
d. The student body is reconsidering the size of the flag. It is now considering enlarging the flag so that it is 3
or 4 times the width of Archie's model. How much would the cloth for a similar flag that is 3 times as wide as
Archie's model cost? What if the flag is 4 times as wide? To answer this question, first estimate how many of
Archie's drawings would fit into each enlarged flag. Then obtain the remaining flags from your teacher.
Confirm each answer by fitting Archie's scale version into the enlarged flags.
e. Complete the table below with your findings. Archie's model is already done for you. What patterns to you
notice? How does the area change when you increase the side lengths?
Linear Scale Factor
1
# of flags
1
Cost
$2
2
3
4
8
15
r
pg.14
8.31 – R:R:R2
Use your pattern you found in the previous problem to answer the following questions.
a. Rectangle ABCD is divided into nine smaller congruent rectangles. The
larger rectangle is similar to the smaller one. What is the linear scale factor?
What is the ratio of their perimeters? What is the ratio of their areas?
b. Kelly's shape at right has an area of 17 mm2. If she enlarges the shape with a
linear scale (zoom) factor of 5, what will be the area of the enlargement? Show
how you got your answer.
c. Examine the two similar shapes at right. What is the linear scale (zoom)
factor? What is the area of the smaller figure?
d. While ordering carpet for his rectangular office, Trish was told by the salesperson that a 16'-by-24' piece of
carpet costs $200. Trish then realized that she read his measurements wrong and that his office is actually 8'by-12'. "Oh, that's no problem," said the salesperson. "That is half the size and will cost $100 instead." Is that
fair? Decide what the price should be. It might be helpful to draw a picture.
8.33 – CONCLUSIONS
How does the area change when you multiply the sides? How does the perimeter change?
pg.15
8.7 – How does the Area Change?_____________________________
Ratios of Similarity
Today you will continue investigating the ratios between similar figures. As you solve today's problems, look
for connections between the ratios of similar figures and what you already know about perimeter and area.
8.34 – TEAM PHOTO
Alice has a 4"-by-5" photo of your school's championship girls gymnastics team. To
celebrate their recent victory, your principal wants Alice to enlarge her photo for a
display case near the main office.
a. When Alice went to the print shop, she was confronted with many choices of sizes:
7"-by-9", 8"-by-10", and 12"-by-16". She is afraid that if she picks the wrong size,
part of the photo will be cut off. Which size should Alice pick and why?
b. The cost of the photo paper to print Alice's 4"-by-5" picture is $0.45. Assuming that the cost per square inch
of photo paper remains constant, how much should it cost to print the enlarged photo?
c. Unbeknownst to her, the Vice-Principal also went out and ordered an enlargement of Alice's photo.
However, the photo paper for his enlargement cost $7.20! What are the dimensions of his photo?
8.35 – PERIMETER AND AREA RATIOS
So far, you have discovered and used the relationship between the areas of similar figures. How are the
perimeters of similar figures related? Confirm your intuition by analyzing the pairs of similar shapes below.
For each pair, calculate the areas and perimeters and complete the table below. Reduce all fractions to lowest
terms.
10
b.
24
pg.16
8.36 – ARE THEY SIMILAR
While Jessie examines the two figures at right, she wonders if they are similar. Decide with
your team if there is enough information to determine if the shapes are similar. Justify your
conclusion.
8.37 – PERIMETER AND AREA RATIOS
Your teacher enlarged the figure at right so that the area of the similar shape is
900 square cm. What is the perimeter of the enlarged figure? Show all work.
8.38 – R:R:R2
How do the ratio of the sides, perimeters, and areas
compare?
8.39 – CONCLUSIONS
Using what you have learned, fill in the chart below.
Side Ratio
2
Perimeter
Ratio
2
Area Ratio
4
12
r
4
20
49
225
pg.17
8.8 – What if the Polygon Has Infinite Sides?_____________________
A Special Ratio
In previous chapters you discovered the perimeter of polygons. Today you will discover how to find the
perimeter of a shape with no sides, a circle.
8.40 – PARTS OF A CIRCLE
Examine the picture of the circle shown. Explain how the radius and diameter are
related. Then explain what the circumference of a circle measures.
8.41 – RATIOS OF CIRCLES
Find the measure of the circumference by wrapping a string around the entire perimeter of the circle and then
measuring how long the string is in centimeters. Then estimate the diameter (length across the circle). Then
write the ratio. What do you notice? What value are you finding?
B
D
A
C
Circle A
Circle B
Circle C
Circle D
Circumference
Diameter
ratio of
circumference
diameter
pg.18
8.42 –CIRCUMFERENCE FORMULA
a. Use the relationship you just discovered to find the formula for the circumference of a circle.
circumference
=
diameter
b. Since the diameter is twice the size of the radius, write a new equation for the
circumference of a circle when given the radius instead of the diameter by substituting in
d = 2r.
8.43 – MISSING INFORMATION
Using your understanding of the relationship between circumference (perimeter of a circle) and the diameter,
find the missing values.
a. Find the circumference of a circle with a diameter of 12 inches.
b. Find the diameter of a circle with a circumference of 28.26 feet.
c. Michael recognized that "pi" is not exactly 3.14. He decides to make sure his answers are precise, so he
doesn't plug in the approximate value of . He leaves the symbol  in his answer. He finds the circumference
of a circle was 22 yards. What is the diameter?
d. Michael wanted to find the exact value of the circumference with a radius of 3 meters. Find the diameter and
then find the exact value. Be sure to have the  symbol in your answer.
pg.19
8.44 – WHEELS GO ROUND AND ROUND
a. Kiley's in-line skate wheels have a 0.42m diameter. How many meters will Kiley
travel after 5000 revolutions of the wheels on her in-line skates?
b. The wheel of one car has a diameter of 20in. The wheel of another car has a diameter of 24in. About how
many more revolutions must the smaller wheel make than the larger wheel to travel 100in?
8.45 – ARC LENGTH
a. Examine the circles below. Find the part of the circumference that is between points A and B by using the
portion of the circle being covered out of 360.
b. A portion of a circle (like the crust of a slice of pizza) is called an arc. This is a set of
connected points a fixed distance from a central point. The length of an arc is a part of the
circle's circumference. If a circle has a radius of 6 cm, find the length of an arc with a
central angle of 30.
pg.20
8.46 – PENDULUM
Find the distance traveled in one back and forth swing by the weight of a 16in pendulum that swings through a
60 angle.
60
16in
8.47 – CONCLUSIONS
What is the perimeter of a circle called? What special value can be found in a circle when you take the
circumference and divide the diameter? How are the radius and the diameter related?
8.48 – FIND THE MISSING PIECES
Using what you have learned, fill in the chart below.
Radius
2
Diameter
4
Circumference
4
12
r
6
30
10
20
pg.21
8.9 – What's the Relationship?_____________________________
Area and Circumferenece of Circles
In the previous lesson, your class discovered that the circumference is 2 units. Today you will develop a
method to find the area of circles. You will also explore parts of circles (called sectors) and learn about their
measurements.
8.49 – AREA OF A CIRCLE
Now that you know the circumference (perimeter) of a circle, how can you find the area of a circle with any
radius?
a. First investigate how the circles are related. Examine the circles at right.
Since circles always have the same shape, what is the relationship between
any two circles?
b. What is the ratio of the circumferences (perimeters)? What is the ratio of
the areas? Explain.
c. If the area of a circle with radius of 1 is  un2, what is the area of a circle with radius 3 units? With radius 10
units? With radius r units?
8.50 – AREA PRACTICE
Based on the formula you developed in the previous problem, answer the questions below.
a. Find the area of a circle with radius 10 units.
b. If the area of a circle is 121 square units, what is the diameter?
c. If the circumference of a circle is 20 units, what is its area?
pg.22
8.51 – PIZZA CELEBRATION
To celebrate their victory, the girls gymnastics team went out for pizza.
a. The captain ate half of a pizza that had a diameter of 20 inches! What was the area
of the pizza she ate? Leave your answers in terms of .
20in
b. As the evening drew to a close, Sonya noticed that there was only one slice of the captain's
pizza remaining. She measured the central angle and found out that it was 45. What is the
area of the remaining slice? Show how you got your answer. Leave your answers in terms of
.
c. Sonya chose a slice from another pizza that had a radius of 8 inches. If her slice had a
central angle of 72, what is the area of this slice? Show how you got your answer.
Leave your answers in terms of .
20in
45
8in
72
d. A region that resembles a slice of pizza is called a sector. It is formed by two radii of a
central angle and the arc between their endpoints on the circle. If a circle has radius 10 feet,
find the area of a sector with a central angle of 20.
pg.23
8.52 – CONCLUSIONS
How do you find the area of a circle?
How does knowing the area of a circle help you
find the area of a sector?
8.53 – FIND THE MISSING PIECES
Using what you have learned, fill in the chart below.
Radius
3
5
Diameter
6
Circumference
6
Area
9
r
14
50
12
400
pg.24
Chapter 8 Closure
What have I learned?
Reflection and Synthesis
8.54 – TOPICS
What have you studied in this chapter? What ideas and words were important in what you learned? Be as
detailed as possible.
8.55– CONNECTIONS
How are the topics, ideas, and words that you learned in previous courses connected to the new ideas in this
chapter? Again, make your list as long as you can.
8.56 – VOCABULARY
The following is a list of vocabulary used in this chapter. Make sure that you are familiar with all of these
words and know what they mean.
Arc length
Circumference
Exterior angle
Non-convex (concave)
Polygon
Sector
Area
Convex
Interior angle
Perimeter
Radius
Similar
Central angle
Diameter
Linear scale factor
Pi ()
Regular polygon
Scale factor
pg.25