ANALYTIC
GEOMETRY
UNIT 3
Circles
Lesson 4
Arc length =
𝜃
360
Area of a sector =
(2𝜋𝑟)
𝜃
360
(𝜋𝑟 2 )
Student Edition
Unit 3 Lesson 4
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Mr. Hastings
CONTENT MAP
Unit 3 – Circles and Spheres - Lesson 4
Essential Questions: How do you use the properties of circles to solve problems involving the
length of an arc and the area of a sector?
INTRODUCTION
Students will continue their study of measurement geometry with a study of length of an arc and
area of a sector.
KEY STANDARDS ADDRESSED
Find arc lengths and areas of sectors of circles
MCC9-12.G.C.5
Derive using similarity the fact that the length of the arc intercepted by an
angle is proportional to the radius, and define the radian measure of the angle as the constant of
proportionality; derive the formula for the area of a sector.
SELECTED TERMS AND SYMBOLS

Arc: an unbroken part of a circle; minor arcs have a measure less than 1800; semi-circles
are arcs that measure exactly 1800; major arcs have a measure greater than 1800

Arc Length: a portion of the circumference of the circle

Arc Measure: The angle that an arc makes at the center of the circle of which it is a part.

Sector: the region bounded by two radii of the circle and their intercepted arc
WORD WALL
Previously Learned Vocabulary
Radius, Diameter, Circuference, Arc
Measure, Minor Arc, Major Arc
Unit 3 Lesson 4
New Vocabulary
Arc Length, Sector
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Analytic Geometry: Unit 3
Essential
Question(s)
Standard(s)
Opening
Lesson 4
How do you use the properties of circles to solve problems involving the
length of an arc and the area of a sector?
MCC9-12.G.C.5
Derive using similarity the fact that the length of the
arc intercepted by an angle is proportional to the radius, and define the
radian measure of the angle as the constant of proportionality; derive the
formula for the area of a sector.
Day One: Do Now: Circumference and Pi Review
Day Three: Do Now: Finding arc lengths of inscribed triangle
Day Five: Do Now: Solving proportions (not included)
Day Six: Do Now: Finding areas found in sectors
.
Work Session
Closing
Unit 3 Lesson 4
Day One: Introduction of unit with cookie task
Day Two:

GO for Length of Arc

Class examples and You Try

Arc Length Practice
Day Three/Four:

Investigating the area of a circle

GO: Area of a sector

Class examples and your try

Practice
Day Five:

Task: Investigating Arc Length and Area of Sectors as
Proportions
Day Six:

Applications of Arc Length and Area of Sectors Practice
Students may choose to use the formula first learned or
proportions to solve (a sample of some problems is included)
Day Two: Ticket out the door (#1)
Day Four: Ticket out the door (#2)
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Do Now: Circumference and Pi Review
𝜋 ≈ 3.14159265358979 …
How do we use Pi to find the circumference of a circle?
Pi is an irrational number. It is a number that can’t be written as the quotient of
two integers. Therefore, Pi will never terminate or repeat. We know that Pi is
found by the quotient of the circumference of the circle and the diameter of the
circle. Therefore, this equation is true:
𝐶
𝜋=
𝑑
By using a basic algebraic operation, we can multiply the d on both sides of the
equation to get:
𝜋𝑑 = 𝐶
Therefore, by knowing that Pi is the quotient of the circumference of a circle and
the diameter of the circle, we have just shown that the circumference of a circle is
equal to Pi times the diameter.
𝐶 = 𝜋𝑑
Or
𝐶 = 2𝜋𝑟
(since we know the diameter is equal to 2r)
(1)
What is circumference?
(2)
What is the formula for circumference?
(3)
What is the circumference of a circle with a radius of 5 inches?
(4)
What is the circumference of a circle with a diameter of 6 yards?
(5)
What is the radius of a circle when the circumference is 36п meters?
(6)
What is the diameter of a circle when the circumference is 100п feet?
Unit 3 Lesson 4
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Mr. Hastings
AREA OF SECTORS AND LENGTH OF ARCS
COOKIE TASK
Part 1: Hands on Activity
Circle LAB
Materials: Construction Paper, Compass, Protractor, Ruler, Scissors. Students have
own rulers, remaining materials are available in bookcase under board.
1. Draw a circle (use radius larger than two (2) inches) on Construction paper with
Compass. Draw a diameter through the center. Cut out the circle. Place a
mark on the edge of the circle. Hold the circle so it is resting on its edge next to
your ruler and line up the mark with the end of the ruler. Carefully roll the circle
along the edge of the ruler until the mark is directly on the bottom of the circle.
Read corresponding measurement from the ruler and record below.
Circumference = _______cm (or inches and fractions i.e. 4 ¾”)
2. Find the measure of the diameter in cm. or inches.
Diameter = ________cm (inches)
3. What is the ratio of the Circumference to the Diameter? Use your calculator.
C
 ______
d
4. The formula for Area of a circle is π(pi) r2; where r = radius of circle
Find the Area of the cookie.
____________cm2
Cut the circle in half on the diameter. Then cut each half of the circle into two
unequal sectors. You will have 4 different pieces of circle. Each piece is a
sector.
5. Using the protractor, find the Angle Measure of each sector’s central angle.
Angle 1 = _______
Angle 3 = _______
Unit 3 Lesson 4
5
Angle 2 = _______
Angle 4 = _______
Mr. Hastings
6. The formula for the length of an arc in a circle is
𝜽
𝑨𝒓𝒄 𝒍𝒆𝒏𝒈𝒕𝒉 = 𝟑𝟔𝟎 (𝟐𝝅𝒓)
where r = radius and θ = central angle
Using the Arc Length formula, find the measure of each sector’s arc length.
Arc Length 1 = ________cm
Arc Length 2 = ________cm
Arc Length 3 = ________cm
Arc Length 4 = ________cm
7. What is the total length of the 4 arcs? _________cm
How does it compare to the circumference of the circlie?
8. The formula for the Area of a Sector is
𝜽
𝑨𝒓𝒆𝒂 𝒐𝒇 𝒔𝒆𝒄𝒕𝒐𝒓 = 𝟑𝟔𝟎 (𝝅𝒓𝟐 )
where r = radius and θ = central angle
Find the Area of each sector.
Area of sector 1 = _________cm2
Area of sector 2 = _________ cm2
Area of sector 3 = _________ cm2
Area of sector 4 = _________ cm2
9. What is the total area of the four sectors? _________ cm 2
How does it compare to the area of the original circle?
10. Explain why the 4 arc lengths should add to the circumference of your circle. If they did
not add to the circumference of your circle, explain why they did not.
11. Explain why the 4 sector areas should add to the area of your circle. If they did not sum
to equal the area, explain why.
Unit 3 Lesson 4
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How do you find the length of an arc?
The formula for circumference is C = ____.
A circle has _____ degrees.
𝒍𝒆𝒏𝒈𝒕𝒉 𝒐𝒇 𝑨𝑩 =
𝒎∠𝑨𝑶𝑩
(𝟐𝝅𝒓)
𝟑𝟔𝟎°
𝜽
OR
𝜽
(𝟐𝝅𝒓)
𝒍𝒆𝒏𝒈𝒕𝒉 𝒐𝒇 𝑨𝑩 =
𝟑𝟔𝟎°
Class example. Give the exact answer and the approximate answer.
Length of RS = ______≈______
Length of MN = ______≈______
E
G
Unit 3 Lesson 4
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20°
15 in
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Length of ABC = ______≈______ Length of EF = ______≈______
You Try!
Find the length of arc AB in terms of 𝜋 and to the hundredths place.
(1)
(2)
(3)
Find the indicated measure. When finding a length measurement, round to the
nearest hundredths place. When finding a degree measurement, round to the
nearest degree.
(4)
Unit 3 Lesson 4
(5)
(6)
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Arc Length
Find arc length in terms of 𝜋 and to the hundredths place. Find degree
measurements to the nearest degree. Find all other measures to the nearest
hundredths place.
Unit 3 Lesson 4
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Unit 3 Lesson 4
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Mr. Hastings