Lesson no. 3 (Arc Length and Area of a Sector

LESSON NO. 3
ARC LENGTH AND AREA
OF A SECTOR

Lesson No. 3 |Arc Length and Area of a Sector
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ENGAGE

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Engagement Activity 1
Linear and Angular measures
Author: Irina Boyadzhiev
Reference: https://www.geogebra.org/m/EazPPkFV
The applet illustrates the linear and angular
measures of central angle in a unit circle.

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Questions:
1.What can you say about the linear measure
of angle? How about angular measure?
2.Is there a relationship between angular and
linear measures of angle?
3.What can you infer about the relationship of
angular and linear measures of angle?

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Learning Guide Card (LGC) # 1
Arc Length
The length of an arc on a circle of radius r is equal
to the radius multiplied by the angle θ subtended by
the arc in radians. Using s to denote arc length we have
s = rθ.
This should actually be intuitive since the arc
length on the unit circle is equivalent to the angle in
radians.

The figure below shows arc length between
points A and B on the circle. Since we are looking
at a length, we always consider the angle θ
subtended by A and B to be positive. (In each of
the next two figures, both and can be moved.)
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Questions:
-What can you say about the length of an arc
on a circle?
-How is the arc length on the unit circle
related to the angle in radians?

Recall that the area of a circle of radius is given by A = π𝑟2
A circular sector is a wedge made of a portion of a
circle based on the central angle θ (in radians) subtended
by an arc on the circle. Since the angle around the entire
circle is 2π radians, we can divide the angle of the sector's
central angle by the angle of the whole circle 2π to
determine the fraction of the circle we are solving for.
Then multiply by the area of the whole circle to derive the
sector area formula.
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Questions:
-What can you say about the area of a circular
sector?
-How do we determine the fraction of the circle we
are solving in area of circular sector?
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EXPLORE

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Explore
The class will be divided into 8 groups (5-6
members). Each group will be given a
problem-based task card to be explored,
answered and presented to the class. Inquiry
questions from the teacher and learners will
be considered during the exploration activity

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Explore
Rubric/Point System of theTask:
0 point – No Answer
1 point – Incorrect Answer/Explanation/Solutions
2 points – Correct Answer but No Explanation/Solutions
3 points – Correct Answer with Explanation/Solutions
4 points – CorrectAnswer/well-Explained/with
Systematic Solution

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Explore
Assigned Role:
Leader – 1 student
Secretary/Recorder – 1 student
Time Keeper – 1
Peacekeeper/Speaker – 1 student
Material Manager – 1-2 students

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Explore
Problem 1 (Group 1 & Group 2): Minute Hand
of a Clock
The minute hand of a clock is 6 inches long.
(a) How far does the tip of the minute hand
move in 15 minutes? (b) How far does it move
in 25 minutes?

Explore
Problem 2 (Group 3 & Group 4): Movement of a
Pendulum
A pendulum swings through an angle of 20° each
second. If the pendulum is 40 inches long, how far
does its tip move each second?
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Explore
Problem 3 (Group 5 & Group 6): Linear Speed v. Angular
speed
Our earlier “obvious” equation s = rθ, relating arc to angle,
also works with measurements of speed. The angular
speed of an spinning object is measured in radians per unit
of time. The linear speed is the speed a particle on the
spinning circle, measure in linear units (feet, meters) per
unit of time.
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Lesson No. 3 | Arc Length and Area of a Sector
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Explore
Problem 3 (Group 5 & Group 6): Linear Speed v.
Angular speed
Suppose a merry-go-round is spinning at 6
revolutions per minute. The radius of the merry-
go-round is 30 feet. How fast is someone traveling
if they are standing at the edge of the merry-go-
round?

Explore
Problem 4 (Group 7 & Group 8): Watering a
Lawn
A water sprinkler sprays water over a
distance of 30 feet while rotating through an
angle of 135°.What area of lawn receives
water?
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EXPLAIN

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Explain
Group Leader/Representative will present
the solutions and answer to the class by
explaining the problem/concept explored
considering the following questions.

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Explain
Guide Questions:
 What is the problem all about?
 What are the given in the problem?
 What are the things did you consider in
solving the given problem?
 What is/are the unknown in the given
problem?

Explain
Guide Questions:
 What method(s) did you use in solving the given
problem?
 How did you solve the given problem using that
method(s)?
 What particular mathematical concept in
trigonometry did you apply to solve the
problem-based task?
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ELABORATE

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Elaborate
Generalization of the Lesson:
- -What is the relationship between linear and
angular measure of arcs?
- What are the steps in solving problems on
arc length and area of a sector?

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Elaborate
Integration of PhilosophicalViews
In this part, the teacher and learners will relate
the terms/content/process learned in the lesson
about arc length and area of a sector in real life
situations/scenario/instances considering the
philosophical views that can be
integrated/associated to the
term(s)/content/process/skills of the lesson.

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Elaborate
Integration of PhilosophicalViews
Questions :
 What are the things/situations/instances that you can
relate with regard to the lesson about arc length and
area of a sector in real-life?
 Considering your philosophical views, how will you
relate the terms/content/process of the lesson in real-
life situations/instances/scenario?

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Elaborate
Sample PhilosophicalViews Integration from theTeacher:
Arc Length and Area of a Sector
Circle and radius are one of the terms
used in this lesson has many real-life
connections. A circle is a line forming a
closed-loop; every point on which is a fixed
distance from a center point.

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Elaborate
Imagine a straight line segment bent
around until its ends join, then arrange that
loop until it is exactly circular - that is, all
points along that line are the same distance
from a center point.

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Elaborate
Unlike other shapes, a circle has a unique
property of being complete. A circle has an
extensive meaning; it represents wholeness,
totality, original perfection, eternity, infinity,
timelessness, self, and all the cyclic
movement.

Elaborate
According to Hermes Trismegistus, God
is a circle whose center is everywhere
and whose circumference is nowhere.
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Elaborate
Circle implies the idea of a movement and
symbolizes the cycle of time - the perpetual motion of
everything that moves like the planet's journey around
the sun and the rhythm of the universe. Many people
believe that if they have God in them, they are complete,
and people who feel complete are stronger and happier

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Elaborate
The distance from the center to any point
of the circle is known as the radius. Each unit
or radius of the circle helps the circle to resist
giving into forces putting pressure on it from
the outside.

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Elaborate
Sample Philosophical Views Integration from the
Teacher: Arc Length and Area of a Sector
Similarly, each of this unit is a person's faith.
Plenty of this strengthens the grip so as not
to be swayed by the evil. Life is a circle
because of the same and continues
progression from birth and growth to decline
and death.

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EVALUATE

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Evaluate
Solve the following problems:
a.The minute hand of a clock is 5 inches long. How
far does the tip of the minute hand move in 30
minutes?
b. An automatic lawn sprinkler sprays up to a
distance of 20 feet while rotating 30 degrees.What
is the area of the sector the sprinkler covers?

Evaluate
Solve the following problems:
c. Find the area of a sector of a circle with central angle of
7𝜋
6
if the
diameter of a circle is 9 cm?
d. A swing has 165° angle of rotation.
i) If the chains of the swing are 6 feet long, what is the length of
the arc that the swing makes? Round your answer to the nearest
tenth.
ii) Describe how the arc length would change if the length of the
chains of the swing were doubled
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Assignment:
Answer the following questions:
1.What are the six trigonometric functions?
2.What is a reference angle?
Reference: DepED Pre-Calculus Learner’s Material, pages 129-131.
-GNDMJR-

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