1 CIRCLE
ES: LESS
SONS
Chris Delany
Lesson I: Introduction
I
to Circles
Objectivess:
(11) Become fam
miliar with circcles and the parrts of a circle
(22) Develop a general
g
interestt for the upcom
ming lesson
Parts of Leesson: (1) Partss of a Circle, (22) Hand Out: Exploring
E
Radiuus and Diametter, (3) Interacttive Game
Teaching Style:
S
Lecture (Part
(
1), Scaffoolding (Part 2, Part 3)
Part 0: Warm Up: Write everything you think you know
k
about circcles. What is thhe definition off a circle? Whaat are
the parts off a circle? Wheere do you see circles in dailyy life?
Share thesee ideas with the class before going
g
into lessson to get a brooader understannding of the toppic of circles,
enlighten students
s
to the ubiquity of cirrcles. The idea that circles shoow up so muchh in the real woorld (architectuure,
shape of a pizza, crop cirrcles, games whhere students sit
s in a circle) will
w serve as motivation for thhe students andd is
ble to relate to..
something they will be ab
Part 1: Paarts of a Circlee
Explain thee parts of a circcle to the studeents, using the following diaggrams.
Origin/Cennter- the centerr of a circle
Diameter- the longest disstance from one end of a circlle to the other (goes through the origin)
Radius- thee distance from
m the origin of the circle to anny point on thee circle (plural, radii).
D
D=2*R,
if D=len
ngth of diametter, R=length of
o radius
Circumfereence- the distan
nce around a circle
(3) The disstance around a circle is calleed the circumfeerence. The diistance across a circle throughh the center is called
the diametter.
is the raatio of the circuumference of a circle to the diameter.
d
Thus, for any circlee, if you dividee the
circumfereence by the diam
meter, you get a value close to
t
. This relaationship is exppressed in the following form
mula:
C/d = . (This
(
is just a brief
b
introductiion to ;
will be discussedd more thorougghly in the nexxt lesson.)
onnecting any two
t points of a circle
Chord- a liine segment co
Secant- a line that passess through any tw
wo points of a circle
Tangent- a line that only touches one pooint of a circle
2 Arc- a curvved line segmeent that is a parrt of the circum
mference of a ciircle
Sector-a frraction of the area of a circle, a portion that is in between two
t radii and an
a arc (like a sllice of a pie)
Segment- a fraction of th
he area of a circcle, a portion thhat is in betweeen a chord andd an arc
Part 2: Haand Out: Explloring Radius//Diameter
Students too work on the following
f
handdout by themseelves for about 10 minutes, thhen discuss theiir findings withh the
person sittiing next to them
m.
1. Draw a radius/diameteer from the givven point.
Here a radius is drawnn
from th
he given point.
a) Draw
w a radius from
m
the giveen point
m
b) Draaw a radius from
each of the given points.
2. How dooes the radius of the circles relate
r
to the raddius of the otheer circles in thee pictures?
a)
b)
3 Hint: You will need to make use of the center point of the circles, so mark them down on paper after you have
drawn any circles. The circles can be bigger or smaller as you choose. The important thing is to determine the
relationship between the radii of the circles.
3. Draw a line and then two points on the line.
Then draw a circle using one point as a center point and the
other point as a marking the radius. In other words, use the
distance between the two points as the radius. Now draw
another circle using the second point as the center point and
the first point as indicating the radius.
Look at the picture.
a) Now draw a line through the points where the two circles intersect. What do you notice?
b) If you connect the two initial points to the point above them where the two circles intersect (cross), what kind of
figure do you get?
Part 3: Interactive Game
Students will take part in this fun and involving game. We will have to move all the desks to the sides of the room,
and the students will sit in a large circle in the middle of the floor. I will have a long piece of rope, and call on
students to “Show me a diameter” or “Show me the circumference” using the piece of rope. Students will be able to
ask others for help if they need it. I will also be explaining definitions of the parts of a circle as we go, and asking
questions like, “So what is the area between the chord and the outside of the circle called? –Segment”
Homework:
Students will have the rest of the time of class to write in their math journals about what they learned or to start
working on their homework (HW 1).
4 Lesson II: Arcs, Chords, Tangents
Objectives:
(1) Develop a further understanding of arcs, chords, tangents
(2) Interact with chords through technology, using Geometer’s Sketchpad
Parts of Lesson: (1) Arcs, (2) Chords, (3) Group Activity, (4)Tangents
Teaching Style: Lecture (Part 1, Part 2, Part 4), Discovery Learning (Part 3)
Part 0: Warm Up: Write down all the parts of a circle (and draw diagrams) that you remember from yesterday’s
lesson. What relationships between the parts can you think of?
As a class, we will go over this briefly, and I will explain that we are going to look more deeply at arcs, chords, and
tangents in today’s lesson. This material will be useful for solving proofs in the future, and, although a bit more
specific than the “general idea,” it is still very important and will be tested upon.
Part 1: Arcs:
An arc is a part of the circumference of a circle and defined as: 'an arc of a circle is the part of the circle between
two points on the circle'. The longer arc is called the major arc while the shorter one is called the minor arc. Arc is
measured in degrees and length. If the measure of minor arc is i.e. the measure of the central angle intercepted by
the minor arc, then the measure of major arc is (360 - ) i.e. the measure of the central angle intercepted by the
minor arc, then the measure of major arc is (360 - ). E.g. if measure of a minor arc is 100 , then major arc is
(360 - 100 ) = 260 .
Types of Arcs:
major arc, minor arc, and semicircle
• If m AOB < 180 , points A and B and the points of circle in the interior of AOB make up minor arc AB,
written as
• Points A and B and the points of the circle not in the interior of AOB make up major arc AB, written
as
• If m
(Fig: 1),
or
AOC = 180 , points A and C separate circle O into two equal parts, each of which is called a semicircle. In
and
two different semicircles
Congruent Arcs:
Central angle is an angle whose vertex is the center of a circle ‘O’. Any central angle intercepts the circle at two
points, thus defines the arc (Fig: 2).
5 Congruent arcs are arcs of the same circle or of congruent circles that are equal in measure. In (Fig: 3, above),
if
O
O' and mCD = mC'D' = 60 , then
Arc Addition Postulate
If AB and BC are two arcs of the same circle having a common endpoint and no other points in common,
then AB + BC =
and mAB + mBC = m
.
Theorems
•
In a circle or in congruent circles, if central angles are congruent, then their intercepted arcs are congruent
(Figure below).
if
O
then
•
•
m
O',
AOB
and
COD, and
AOB
A'O'B'.
In a circle or in congruent circles, central angles are congruent if their intercepted arcs are congruent.
In a circle or in congruent circles, central angles are congruent if and only if their intercepted arcs are
congruent.
E.g. (In figure below) P, Q, S, and R are points on circle O, mPOQ = 100 ,m QOS = 110 , and m
SOR = 35 . Find m ROP.
ROP will be- [360 – (100 + 110 + 35 )] = 115 , as the final answer.
6 Part 2: Chords:
A chord of a circle is a line segment whose endpoints are points on the circle. E.g. in the Fig: 4, AB and AOC are
chords of circle O. Thus, a diameter is a special chord of a circle that has the center of the circle as one of its points.
In the Fig: 4, AOC is the diameter.
Important Properties of chords
• every chord defines an arc whose endpoints are the same as those of the chord and AD = DB. E.g. diameter and
semicircle are the chord and arc that share the same endpoints.
• the central angle forms an isosceles triangle, with chord as one side and the other two sides are rays that make the
central angle.
• the only diameter perpendicular to the given chord, is the perpendicular bisector also of that chord.
Part 3: Group Activity
Students will break into groups of two or three and work on the group activity entitled ‘Chords in a Circle’ for
Geometer’s Sketchpad. The use of Geometer’s Sketchpad will allow students to discover relations and conclusions
on their own, being able to interact with the concepts with a more hands-on approach.
7 8 9 Part 3: Tangents:
A tangent to a circle is a line in plane of the circle that intersects the circle in one and only one point (Fig: 18).
A secant of a circle is a line that intersects the circle in two points.
Postulate
At a given point on a given circle, one and only one line can be drawn that is tangent to the circle.
Theorems:
• if a line is perpendicular to a radius at a point on the circle, then the line is tangent to the circle.
• if a line is tangent to a circle, then it is perpendicular to the radius at a point on the circle.
Common Tangents
A line that is tangent to each of two circles is called a common tangent.
Homework:
HW 2: Display your understanding of 5 of the following by the use of pictures, diagrams, and/or proofs:
-Congruent Arcs, Arc Addition Postulate, The 3 Arc Theorems, Important Properties of Chords, Tangent Postulate,
The 2 Tangent Theorems, Common Tangents
10 Lesson III: Discovering Pi
Objective: (1) To get the students more interested in the idea of pi and this lesson plan about circles.
(2) To get the students to understand the concept of pi, how it is a ratio comparing diameter and
circumference
Overview: (1) “Discovering Pi”, (2) Some Problems, (3) Historical Information about Pi, (4) More Pi Fun
Teaching Style: Scaffolding/Cooperative Learning (Part 1), Lecture -with class involvement (Parts 2, 3, 4)
Part 1:
Discovering Pi
AUTHOR: Jack Eckley, Sunset Elem., Cody, WY
Date: 1994
Subject(s): Mathematics/Geometry
OVERVIEW: Many students tend to memorize, without understanding, formulas that we use in geometry or other
mathematic areas. This particular activity allows students to discover why pi works in solving problems dealing with
finding circumference.
OBJECTIVES: The students will:
1.
2.
3.
4.
5.
Measure the circumference of an object to the nearest millimeter.
Measure the diameter of an object to the nearest millimeter.
Explain how the number 3.14 for pi was determined.
Demonstrate that by dividing the circumference of an object by its diameter you end up with pi.
Discover the formula for finding circumference using pi, and demonstrate it.
RESOURCES/MATERIALS:
•
•
•
•
•
round objects such as jars, lids, etc.
measuring tapes, or string and rulers
paper
pencil
calculator
ACTIVITIES AND PROCEDURES:
1.
2.
3.
4.
5.
6.
Divide class into groups of three or four.
Give materials to student teams.
Have student teams make a table or chart that shows name or number of object, circumference, diameter,
and ?.
Have students measure and record each object's circumference and diameter, then divide the circumference
by the diameter and record result in the ? column.
Have students find the average for the ? column and compare to other groups in the class to determine a
pattern. Students can then find the average number for the class.
Explain to the students that they have just discovered pi, which is very important in finding the
circumference of an object. (You may wish to give some historical information about pi at this time or have
students research the information.)
11 7.
Have students come up with a formula to find the circumference of an object knowing only the diameter of
that object, and the number that represents pi. Students must prove their formula works by demonstration
and measuring to check their results.
TYING IT ALL TOGETHER:
1.
2.
3.
Have students write their conclusions for the activities they have just done. Students may also share what
they have learned with other members of the class.
Give students three problems listing only the diameter of each object and have them find the circumference.
Encourage students to share learned knowledge with parents
Part 2: Some Problems
Say, “As we just learned, pi is a constant value that describes the ratio of a circle’s diameter to its circumference.
We see that C/d =
or C = d. Using this information solve for the circumferences of the circles with the given
information: (solve both keeping
in your answer, and with using =3.14”
1. D=5 cm
2. D=3 in
3. R=.5 cm
(Have the students work on these problems by themselves, then check with the person sitting next to them. Then, we
will go over them as a class.)
Say, “ is used in several other formulas, such as area of a circle, area of a sector, volume of a sphere, surface area
of a cylinder, etc. We will be learning about some of these other formulas (area of a circle, area of a sector,
circumference, length of an arc) in the next few days.”
12 Part 3: Historical Information about Pi
Just for fun, I will give the students the following information to get them more involved in the idea of pi.
Facts about Pi
, a fundamental constant of nature, is one of the most famous and most remarkable numbers
you have ever met.
The Egyptians and the Babylonians are the first cultures that discovered
about 4,000 years
ago. Here is a small table that shows some of the very old discoveries of :
Culture/Person
Approximate Time
Value Used
Babylonians
2000 BC
3 + 1/8 = 3.125
Egyptians
2000 BC
3.16045
China
1200 BC
3
Bible mentions it
550 BC
3
Archimedes
250 BC
3.1418
Hon Han Shu
130
sqrt (10) = 3.1622
Ptolemy
150
3.14166
William Jones, a self-taught English mathematician born in Wales, is the one who selected the
Greek letter
for the ratio of a circle's circumference to its diameter in 1706.
is an irrational number. That means that it can not be written as the ratio of two integer
numbers. For example, the ratio 22/7 is a popular one used for
but it is only an
approximation which equals about 3.142857143...
Another more precise ratio is 355/113 which results in 3.14159292... This was given to me by
a student.
Another characteristic of
as an irrational number is the fact that it takes an infinite number
of digits to give its exact value, i.e. you can never get to the end of it.
Since 4,000 years ago and up until this very day, people have been trying to get more and more
accurate values for pi. Presently supercomputers are used to find the value of
with as many
digits as possible. Pi has been calculated with a precision containing more than one billion
digits, i.e., more that 1,000,000,000 digits!
Here are three different ways to approximate the value of :
1.
2.
3.
/ 2 ~= (2*2*4*4*6*6*8*8*...) / (1*3*3*5*5*7*7*9*...)
/ 4 ~= 1 -1/3 + 1/5 - 1/7 + 1/9 - 1/11 + ...
~= 3 + 1/10 + 4/102 + 1/103 + 5/104 + 9/105 + ...
The symbol "~=" means approximate. They are not equalities but can be very close.
Try them out in your calculator! It is fun!
13 Part 4: More Pi Fun
On an projection screen, I will pull up the following websites and go through them with the class. The first website
is where you can find the digits of your birthday in pi (as well as its location). I will select several students in the
class and put their birthdays into the website, and the class can watch. The second website shows clever T-shirts
designed with images of pi. I think that both of these websites will engage the students further into the lesson and
get them more interested in the idea of pi. At the end of the class, I will write the links on the board to make them
available for the students to use at home.
http://www.angio.net/pi/piquery is a website where you can find your birthday in pi.
http://www.zazzle.com/pi+tshirts is a website showing funny pi T-shirts. (“pumpkin pi”, “cow pi”, “cutie pi”)
Homework:
HW 3: Continue investing pi online. Find one interesting pi fact, and we will share them as a class tomorrow. For
example, you can read about the world record holder for the most digits of pi memorized. Write down your pi facts
in your math journal.
http://www.pi-world-ranking-list.com/lists/details/luchaointerview.html
14 Lesson IV: Measuring Ciircles
Objectivess:
(11) Explore conccepts of and diifferent ways of
o measuring paarts of a circle in relationshipp to different ciircle
paarts
(22) Learn relatio
onship betweenn degrees and radians
r
Parts of Leesson: (1) Unitss, (2) Circumfeerence/Area, (33) Arcs, (4) Sectors, Segmentts
Teaching Style:
S
Lecture (Part
(
2), Lecturre/Cooperativee Learning (Parrt 1), (Part 3), (Part
(
4)
Part 0: Warm Up:
As a class, students will share
s
their “pi facts” from lasst night’s homeework. This shhould get them interested and
awake, whhich will be goo
od because thiss lesson will bee a good bit of lecture.
nits
Part 1: Un
A degree iss defined as 1/360 of a rotatioon of a radius about
a
the centeer of the circle.. Simply put, a circle is divided
into 360 eqqual degrees, and
a a right anglle (1/4th of the rotation) is 900. The degree measure
m
dividees a circle into 360
equal partss,
Another unnit of measure is the gradient. A gradient breaks
b
a circle into
i
400 equal parts i.e. theree are 100 gradieents
in a right angle.
a
The use of
o a gradient makes
m
measurem
ment “fit better” with the deccimals system.
Another method
m
for meassuring angles in case of the circle is radian
n measure.
A radian iss the measure of
o an angle of an
a arc with its length
l
= to thee radius.
ns in a compleete circle (ask class:
c
Why is thhis?)
Thus, theree are 2 radian
The measuure in radians can
c be determinned by the interrsected arc lenngth (s) dividedd by radius (r) and it can be
expressed as:
(radianss) = arc length//radius = s/r
Radians usseful especially
y in applicationns of calculus involving
i
trigonometric functtions— (sine, cosine,
c
tangentt, etc.)
In such casses the angle of the trigonomeetric function be
b measured inn radians.
- Degrees and
a radians aree the two units of measuremeent we will use in this class. These
T
two unitss are very easyy to
convert bettween. Keep in
n mind that 2 radians = 360 and
raddians = 180 .
-Ask class,, “How can wee develop a form
mula to converrt between deggrees and radianns (and vice veersa)? Discuss this
with the peerson sitting neext to you.”
-Then, go over
o
solution with
w the class, explaining
e
in more
m
detail.
We can sollve this by settting up an equaation. We know
w that 360 =2 radians, so we
w can solve 360
3
(factor)=22
radians forr the multiplyin
ng factor and get
g
/180 . To
T get from deggrees to radianss, we will multtiply by /180 .
The radianns to degrees co
onversion is sim
milar—to get from
f
radians too degrees, we simply multiplyy by 180 / .
A
Part 2: Reecap: Circumfference and Area:
Formulas for
f circumferen
nce and area will
w be useful inn the following ways to measuure a circle (arrc length, area of
o a
sector, etc)).
15 You may recall from earlier that the ratio of the circumference of a circle to the length of its diameter (d), is
see that the formula for circumference is C = d, or (since we know d = 2r) , C = 2 r.
The formula for area of a circle is A = r^2
. We
Basic Problems (to be done on board):
Find Circumference:
1. D= 2 cm
2. R=4 cm
Find Area
1. D=2 cm
2. R=4 cm
Part 3: Arcs:
Degree Measure of an Arc: An arc of a circle is the part of the circle between two points on the circle. An arc of a
circle is called an intercepted arc, or an arc intercepted by an angle, if each end point of the arc is on a different ray
of the angle and the other points of the arc are in the interior of the angle. The degree measure of an arc is equal to
the measure of the central angle that intercepts the arc. E.g.
degree measure of an arc can be anywhere from 0 to 360 .
. There are 360 total in a circle, so the
Length of an Arc:
We see that an arc is a fraction of the whole circumference of the circle. Thus, if
is the measure of central angle,
then the length of the arc intercepted by the angle is given by
/ 360 * 2 r. For example, if
= 90 and
r=6, arc length would be (1/4)*2* *6 = (1/4)(12) =3 .
Example Problems: (students will have time to do these on their own, then we will discuss as a class)
Find the arc lengths of circles with the following information.
=45, r = 10 cm
2. =60, r = 8 in.
3. =110, d= 7 m.
1.
Part 4: Sectors:
A sector of a circle is a pie shaped portion of the circle area, and it is between two segments coming out of the center
of the circle (Fig: 16). Another way to say, it is the region enclosed by the central angle of a circle and the circle
itself. A segment of a circle is the region enclosed by a chord and the arc that the chord defines (Fig: 17). The figure
given below will help you understand it easily:
For example: Find the area of the segment of a circle in Fig: 17,
16 To studentts: Now that yo
ou know how too solve for lenggth of an arc, hypothesize
h
in groups of 3 orr 4 on how to fiind
the areas of
o the sector and
d segment in thhe figures abovve.
blem in class, and
a I can explaain how we find the area of a sector (
We will goo over this prob
the area off a segment (areea of sector – area
a of the equil. triangle).
Solution too the problem:
Given: the central angle of
o the segmentt is 60 and the radius is 8cm
m.
( / 6)*( r2) = (1 / 6)*( * 82) = (1 / 6) * [
Step 1: Areea of sector = (1
Step 2: Areea of triangle OAB
O = (82 3) / 4 = 27.71 cm
m2
Step 3: Areea of segment = (33.49 – 27..71)
= 5.78 cm2
Area of seggment is 5.78 cm
c 2, as the finaal answer.
/ 3660 *
*r^2)) and
* (64)] = 33.49 cm2
In summarry:
Length of an
a arc= / 360 * 2 r
Area of a sector=
s
/ 360
0 * *r^2
Area of a segment=
s
area of sector – areea of equilateraal triangle
P
(stud
dents will have time to do theese on their ownn, then we willl discuss as a class)
c
Example Problems:
Find the arreas of the secttors and segments formed (if an equil. trianggle was to be made)
m
of circles with the folloowing
informationn.
1.
= r = 10 cm
=45,
3. =110, d= 7 m.
2. =60, r = 8 in.
Homework:
Students will
w have the reest of time to sttart their homeework (HW 4) or ask the teacher if they havve any questionns
about whatt they learned that
t day.
17 Lesson V: Using a Compass
Objectives:
(1) Introduction to compasses and how to use them
(2) Obtain a better understanding of compasses, using them to construct circles with 3 points and angle
bisectors
Parts of Lesson: (1) Introduction to compasses, (2) Group work: compass practice and activities
Teaching Style: Lecture (Part 1), Cooperative Learning/Group work (Part 2)
Part 0: Warm Up
For the first part of class, I will give some time for students to just play around with the compasses and see if they
can figure out how to use them on their own. They will be able to talk to the people sitting around them (about
compasses) if they wish (this will be a discovery/cooperative learning style of teaching).
Part 1: Introduction to compasses
First, I will provide a lesson on what a compass is and how to properly use it to draw circles. A compass looks like
this:
Explain these directions:
We can use an instrument called a compass to draw circles. To use a compass, fasten a pencil in the pencil hold and
adjust the hinge so that the distance between the compass needed and the pencil tip is the desired radius length.
Then, put the compass needle on a piece of paper where you want the origin to be, bring the pencil tip so that it is
touching the paper, and rotate the pencil around the origin until a complete circle is made. We see that all the points
we have drawn are equidistant from the origin (so, by definition, we have a circle).
Tell the students to practice drawing circles of various radii or other qualifications.. one with radius 3 cm, one with
radius 4cm, one with diameter 5 cm, one with circumference 10 cm.
Part 2: Group Work
The students will then break into groups and work on the following two assignments, discovering how to construct a
circle through 3 points and how to find the center of a circle using a compass. In order to do these activities, the
students will also have to learn how to construct a perpendicular bisector (information attached). The extra practice
using a compass will be beneficial for students, and hopefully they will increase the students’ increase in the subject
matter.
18 (1) Constructing a Circle through 3 points
After doing this
We start with three given points. We will construct a circle that
passes through all three.
1. (Optional*) Draw straight lines to create the line segments AB
and BC. Any two pairs of the points will work.
* We draw the two lines to make it clear when we later draw their
perpendicular bisectors, but it is not strictly necessary for them to
actually be there to do this.
2. Find the perpendicular bisector of one of the lines. See
“Constructing the Perpendicular Bisector of a Line Segment.”
Your work should look like this
19 After doing this
3. Repeat for the other line.
4. The point where these two perpendiculars intersect is the center
of the circle we desire.
5. Place the compass point on the intersection of the
perpendiculars and set the compass width to one of the points A,B
or C. Draw a circle that will pass through all three.
Your work should look like this
20 After doing this
Your work should look like this
6. Done. The circle drawn is the only circle that will pass through
all three points.
(2) Finding the center of a circle
After doing this
We start with a given circle.
Your work should look like this
21 After doing this
1. Using a straightedge, draw any two chords of the circle.
For greatest accuracy, avoid chords that are nearly parallel.
2. Construct the perpendicular bisector of one of the chords
using the method described in “Constructing a perpendicular
bisector of a line segment.”
3. Repeat for the other chord
Your work should look like this
22 After doing this
Your work should look like this
4. The point where the two lines intersect is the center C of
the circle.
Constructing the perpendicular bisector of a line segment: This will be necessary to complete the other two
activities.
After doing this
Start with a line segment PQ.
1. Place the compass on one end
of the line segment.
2. Set the compass width to a
approximately two thirds the line
length. The actual width does not
matter.
3. Without changing the compass
width, draw an arc on each side of
the line.
Your work should look like this
23 After doing this
Your work should look like this
4. Again without changing the
compass width, place the compass
point on the the other end of the
line. Draw an arc on each side of
the line so that the arcs cross the
first two.
5. Using a straightedge, draw a
line between the points where the
arcs intersect.
6. Done. This line is perpendicular
to the first line and bisects it (cuts it
at the exact midpoint of the line).
Homework:
HW 5: Write about the process of how to use a compass in your math journal. What did you learn from today’s
activities? How could you use a compass in a real-world application? What would be another tool that could be used
to draw circles?
24 Lesson VI: Standard Form
m, Graphing Circles
C
Objectivess:
(11) Understand how
h to obtain and
a use the staandard form off the equation of
o a circle
(22) Explore how
w to convert bettween the standdard and generral form of a ciircle
(33) Know how to
t write an equation in standaard form given either the radiuus and center or
o by looking at
a a
grraph
(44) Introduction
n to use of ‘com
mpleting the squuare’ to go from
m general to sttandard form
Parts of Leesson: (1) Stand
dard and Geneeral form, (2) Group
G
Work
Teaching Style:
S
Lecture (Part
(
1), Coopeerative Learninng (Part 2)
G
forms of a circle
Part 1: Staandard and General
The standaard form of a circle is (x - h))2 + (y - k)2 = r2, where (h, k) is the center and
a r is the radiius of a circle. Also,
when the center
c
of the cirrcle is (0, 0), annd the radius iss r, then the eqquation of a circcle is x2+ y2 = r2. Thus if
the radius of
o a circle is 2 with the center at origin, the equation of thhe circle is x2 + y2 = 4. Every circle centeredd at
the origin will
w have this type
t
of equatioon.
The general form of a ciircle will be off form x2 + y2 + ax + by + c = 0. We will leearn how to connvert between
general andd standard form
ms soon.
Examples: finding the co
oordinates of thhe center and thhe radius of a circle
c
from equuation of a circlle.
Ex 1. Find the center and
d radius of the circle
c
whose eqquation is (x - 5)2 + (y - 1)2 = 36.
By remembbering the stan
ndard form as (x
( - h)2 + (y - k)
k 2 = r2, we cann solve for h, k,, and r, such that h = 5, k = 1 and r
= 36= 6. Thus, center of
o the circle is (5,
( 1) and r = 6.
6
Ex 2. If thee center of circcle is (3, -1) andd radius 7; to find
f
the equatioon of the circlee you plug in thhese values in the
t
equation inn standard form
m (x - h)2 + (y - k)2 = r2 and itt results in the equation i.e. (xx - 3)2 + (y + 1)2 = 49.
Now do soome problems on
o your own:
Solve for the standard an
nd general form
m of a circle givven the followiing informationn.
1. Center
3. x^2+y^2+
C
(5, -4), r = 4
2. x^2+4x+44 + y^2+6y+9 = 0
+6y+1 = 0
We will then go over these problems ass a class. Problem number 3 requires
r
a “com
mpleting the sqquares” approacch,
but we will see if any of the
t students wiill be able to fiigure it out on their
t
own (disccovery methodd). Then, I will teach
the entire class
c
about com
mpleting the sqquare.
ng the square::
Completin
Given the equation
e
x2 + y2 - 4x + 6y +44 =0 into the staandard form foor the circle, annd find its radiuus.
You can foollow the steps given below:
The standaard form of a ciircle is (x - h)2 + (y - b)2= r2, where (h, k) iss the center andd r is the radiuss of a circle.
Notice here the term x2 - 4x; and half off -4 is -2. Thenn (-2)2 = 4
Next pick up
u x2 - 4x + 4 and
a factor, thatt results into (xx - 2)2.
2
Now you are
a left with y + 6y. Then 6 / 3 = 2, and (3)2 = 9.
Add 9 to both the sides, and
a that resultss into (x - 2)2 + y2 + 6y + 9 = 9.
2
Note that y + 6y + 9 = (y
y + 3)2 and 9 = 32, so the equaation is (x - 2)2 + (y + 3)2 = 32.
Identify the circle now: It is the circle with
w center at (22, -3) and radiuus 3.
Further if you
y want to draaw this circle, use a compass. On the graphh sheet, mark thhe center and use
u compass to draw
a circle witth 3 unit radiuss.
Completing the square taakes more thinkking than the reest of the thinggs the class will learn, so I wiill try to emphaasize
this conceppt and provide many completting the square questions in thhe homeworks.
Practice Prroblems (some time to be givven in class, thee rest for homeework)
Complete the
t square giveen the followinng equations, soolving for the standard
s
form of a circle and thus being ablle to
give the cirrcle’s center an
nd radius.
4. 4x2 + 4yy2 – 16x – 24y + 51 = 0
1.
.
25 2.
3.
3. 100x2 + 100y2 – 100x + 240y – 56 = 0.
x2 + y2 +6x – 4y = 12
5. 4x2 + 4y2 – 4x + 8y = 11
Graphing Circles:
Once we solve for the center and radius of our circle, we can easily plot them on a graph. Simply plot the center
point of a circle, and then use your compass to draw a circle with the appropriate radius.
Part 2: Group work problems:
The students will split up and work on the following problems, developing their understanding of the subject matter
and learning how to solve these types of problems.
Example 1: Look at the figure below: The circle is located at (-1, -2) and the radius is 1.5 units. What is the equation
of the circle in standard form?
Example 2: Now look at a different case where equation of the circle is not in standard form.
Given: Find the center and radius of the circle, x2 + y2 + 6y + 8 = 0, and graph it.
Hint: Write the equation of the circle in standard form (x - h)2 + (y - k)2 = r2 using the ‘completing the square’
method.
Example 3: Given- the circle equation is, (x + 3)2 + (y - 4)2 = 16. Find out, if the point A (4, 5) is inside, outside or
on the circle.
After the students solve the problems, we will discuss them as a class. I, as the teacher, can work out problems that
need special attention on the board for everyone to see. Example 3 will probably need attention, and I can explain
the steps (find the radius of the circle, find the distance between the center of the circle and pt A and determine if
this distance is less than, equal to, or greater than the radius)
Homework:
With any remaining time in class, students will be asked to write in their math journals what they have learned
today, and then to begin working on their homework (HW 6).
26 Lesson VII: Circumscribed Polygons and Inscribed Angles
Objectives:
(1) Understand and be able to differentiate between inscribed and circumscribed polygons
(2) With past knowledge, come up with ideas on how to find areas of such figures
(3) Explore inscribed angles further with the online activity
Parts of Lesson: (1) Inscribed and Circumscribed Polygons, (2) Group Work, (3) Online Group Activity
Teaching Style: Lecture (Part 1), Cooperative Learning (Part 2, Part 3), Discovery (Part 3)
Part 0: Go over yesterday’s homework, work out any specific problems that students had trouble with. Especially
give attention to the completing the squares problems.
Part 1: Inscribed and Circumscribed Polygons
Polygons Inscribed in a Circle
If all of the vertices of a polygon are points of a circle, then the polygon is said to be inscribed in the circle. In other
words, it can also be expressed that the circle is circumscribed about the polygon.
As you may recall from earlier geometry, a polygon is a closed plane figure bounded by straight line segments as
sides. A regular polygon may be a polygon which is equiangular (all angles are equal in measure) or equilateral (all
sides have the same length). We will see that every regular polygon has an inscribed circle. Examples of the simplest
regular polygons are the equilateral triangle, the square, the regular pentagon etc.
An inscribed regular polygon is a polygon placed inside a circle such that each vertex of the polygon touches the
circle and each of its sides is a chord.
A circumscribed regular polygon is a polygon whose segments are tangent to a circle.
Memory tricks:
An INscribed regular polygon is a polygon INside a circle. (Fig. 2)
A circumscribed regular polygon goes AROUND a circle. (“circum” means “around”.. think of the word
“circumference”) (Fig. 3)
Inscribed Angle:
An inscribed angle of a circle is an angle whose vertex is on the circle and whose sides contain chords of the circle
(Fig: 5).
27 Thus, we see
s that in Fig 2,
2 angles DAB
B, ABC, BCD, CDA
C
are all innscribed angless.
Part 2: Grroup Work
The studennts will split up
p into groups of three or four and work on thhe following problems. We will
w then work out
the problem
ms as a class on
n the board, annd I will be ablle to help as necessary.
Example 1:
1 If you are ask
ked to find outt if it is possiblle to cut out a square
s
with a side 30 cm from
m a circle with a
radius of 20
2 cm.
Hint: You will need to deetermine the reelationship betw
ween length off a side of the square
s
and radiii of the circle.
(This probllem will surely
y need to be woorked out on thhe board.
Solution:
To solve itt, the steps are as follows:
Recall from
m above learnin
ng that the biggest square thaat is included inn a circle, is ann inscribed squuare. Now take help
of the form
mula:
The relatioon between sidees and radii of regular polygoon is: a = 2 * radius = 1.1441 r
Since radiuus here is 20 cm
m, side equals to 1.141 * 20 = 28 cm approxx.
Therefore, it is possible to
t cut a square with side 30 cm
m from a circlee of 20 cm radius or 40 cm diameter, as thee final
answer.)
Example 2:
2 Areas of insccribed and circcumscribed pollygons.
The TA-CO
O method is a method to findd areas of inscrribed and circumscribed polygons. Basicallyy, we will subttract
the cut out (CO) area from
m the total areaa (TA) to get thhe required areea.
Using this knowledge, so
olve the following problem:
If a square floor of side 6m
6 is covered by
b a circular shhape rug (the sqquare is a circuumscribed reguular polygon), find
f
the area off the floor that is
i uncovered by
b the rug.
(Group disscussion will heelp prompt students’ thinkingg and get them to figure out how
h to solve thhe problem as a
group. This problem willl probably be discussed
d
as a class
c
as well. I can draw a piccture of what’s going on (andd
explain thaat this is a good
d strategy for solving
s
such prroblems), explaain the steps, annd go through the solution.
The solutioon is 62 - (3))2 = 27 m2.
Example 3:
3 Identify two polygons inscrribed in the cirrcle.
28 (This problem can be discussed in front of the class as well. I can discuss the following steps to go about this
problem and the rationale to coming to the answer.
Step 1: In case of polygon inscribed in a circle, each of its sides is a chord.
Step 2: For SQP, SQR, SRP, QRP and Quadrilateral SPQR, each side is a chord of the circle.
Step 3: For SRO, SPO, QPO, QRO, some of the sides are not the chords of the circle.
Step 4: Therefore, SQP, SQR, SRP, QRP and Quadrilateral SRQP are the inscribed polygons. )
Inscribed Angle Theorem:
The measure of an inscribed angle of a circle is equal to one-half the measure of its intercepted arc.
Proof If one of the sides of the inscribed angle contains a diameter of the circle.
Consider first an inscribed angle (Fig: 6),
m OAB = m OBA = x
=> m AOC = x + x = 2x
Also, m AOC = m
= 2x
=> m ABC = x = ½
m
ABC, with
diameter of circle O.
Example 4 : find the measure of arc
in Fig: 7.
By the theorem stated above, A and C are supplementary. Therefore, C equals 95 .
(Solution: From the theorem, measure of an arc is double that of its inscribed angle. Therefore, arc
as the final answer. )
190 ,
29 Part 3: Online Group Activity
Group activity:
This activity uses a java applet that allows students to play with parts of a circle (changing radius, points, etc) and
further investigate the idea of inscribed angles. “Discovery” questions are the provided for the students to investigate
(“What happens when this happens?”, etc)
http://www.analyzemath.com/Geometry/CentralInscribedAngle/CentralInscribedAngle.html
Homework:
With any extra time, students will be asked to record their feelings about today’s lesson and then start working on
their homework (HW 7).