Arc length = θ360

(2πr )

Area of a sector = θ360

(π r2 )

Lesson 4

ANALYTIC GEOMETRY

UNIT 3Circles

Student EditionUnit 3 Lesson 4 1 Mr. Hastings

Previously Learned VocabularyRadius, Diameter, Circuference, Arc Measure, Minor Arc, Major Arc

New VocabularyArc Length, Sector

CONTENT MAPUnit 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

Unit 3 Lesson 4 2 Mr. Hastings

Analytic Geometry: Unit 3 Lesson 4

Essential Question(s)

How do you use the properties of circles to solve problems involving the length of an arc and the area of a sector?

Standard(s)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.

Opening Day One: Do Now: Circumference and Pi ReviewDay Three: Do Now: Finding arc lengths of inscribed triangleDay Five: Do Now: Solving proportions (not included)Day Six: Do Now: Finding areas found in sectors

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Work Session Day One: Introduction of unit with cookie taskDay 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

ProportionsDay Six:

Applications of Arc Length and Area of Sectors PracticeStudents may choose to use the formula first learned or proportions to solve (a sample of some problems is included)

Closing Day Two: Ticket out the door (#1)Day Four: Ticket out the door (#2)

Unit 3 Lesson 4 3 Mr. Hastings

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:

π=Cd

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=πdOr

C=2π r

(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 4 Mr. Hastings

Unit 3 Lesson 4 5 Mr. Hastings

AREA OF SECTORS AND LENGTH OF ARCSCOOKIE 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.

Cd

=______

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 2 = _______

Angle 3 = _______ Angle 4 = _______

Unit 3 Lesson 4 6 Mr. Hastings

6. The formula for the length of an arc in a circle is

Arclength= θ360

(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

Areaof sector= θ360

(πr 2)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? _________ cm2

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.

20°

15 in

E

F G

How do you find the length of an arc?The formula for circumference is C = ____.

A circle has _____ degrees.

Class example. Give the exact answer and the approximate answer.

Length of RS = ______≈______ Length of MN = ______≈______

Unit 3 Lesson 4 7 Mr. Hastings

lengthof AB=m∠AOB360°

(2πr )

OR

lengthof AB= θ360 °

(2 πr )

θ

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) (5) (6)

Unit 3 Lesson 4 8 Mr. Hastings

Arc LengthFind 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 9 Mr. Hastings

Unit 3 Lesson 4 10 Mr. Hastings