Section 3.3
Circles
Pre-Activity
Preparation
Cultural Heritage and the Circle
The word mandala originates from the Sanskrit word for
circle. A mandala can be any form of circular geometric
design that contains symbols of a person’s inner self,
guiding principles, and overall ideas about the world.
A mandala symbolizes an imaginary place that is
contemplated during meditation in both Hinduism and
Buddhism. The significance of objects within a mandala
is conveyed by shape, size, and color; these objects can
be abstract designs or specific drawings of people, places,
and ideas that are central to a person’s life.
A mandala is a symbol of wholeness in religious art, the circle of eternity. Mandalas are found in nature. Every
cell in our body, for instance, is a living mandala. So is the iris of our eye, a snow crystal, a bird’s nest―even
a bicycle wheel. Below are some sites to learn more about mandalas in art and religion.
•
The Mandala Project (www.mandalaproject.org) is a non-profit project dedicated to promoting peace
through art and education. The Mandala Gallery contains mandala art from people all over the world.
•
Earthmeasure (www.earthmeasure.com) includes many Native American applications of geometry.
Learning Objectives
• Learn how to calculate the circumference of a circle given its diameter or radius
• Learn how to calculate area of a circle given its radius or diameter
• Distinguish between an exact answer using pi (�) and an approximate answer using a decimal
representation for pi (�)
Terminology
Previously Used
New Terms
to
Learn
area
circumference
irrational number
diameter
perimeter
linear units
radius
pi (p)
195
Chapter 3 — Geometry
196
Building Mathematical Language
Circles
A circle is a basic geometric shape that is the collection of points equidistant (at equal distances) from the
center (point O) by a distance r.
O
d
C
r
Center
O the center point
Radius
r
Diameter
d distance across the middle of the circle (d=2r)
Circumference
C perimeter; the distance around the circle
distance from the center to the edge
Pi
One irrational number that plays an important role in the study of circles is the number pi (π). You
can get close to the value of pi by dividing the circumference, C, by the diameter, d, of any circle and
carrying out the division to the desired number of decimal places. Pi is an exact value, but in calculations it
is often approximated. Decimal approximations of pi have been calculated to more than a trillion decimal
places; the first fifty are: 3.14159265358979323846264338327950288419716939937510...
For our purposes, 3.14 will enable us to have an approximate answer that is as accurate as we need it to be.
π is the ratio of the circumference to the diameter of
a circle. This holds true for every circle, regardless
of the actual measurement of its circumference
and diameter; the ratio is always pi.
π =
C
. 3.14
d
Circumference
The distance around a circle (the perimeter) is the
circumference (the outline of the circle in the figure above).
You can calculate the circumference by using either the diameter
or the radius. As with perimeter, the units for circumference are
linear units (units which measure length: inches, feet, meters,
etc.)
Circumference
C = πd
Area
Calculate the area of a circle by multiplying pi by the square
of the radius. If diameter is given, use the relationship d = 2r or
r = ½ d. The units for area are square units (square feet, square
meters, etc.).
C = 2πr
and
Area
A = πr 2
and
d
A = π 
 2 
(d = 2r, r = ½ d)
2
Section 3.3 — Circles
197
Validate answers by 1) checking for correct linear or squared units and 2) checking arithmetic calculations.
Another good validation technique is to use the alternate measurement: if r is given, use d to validate and
if d is given, use r to validate.
When diameter is given:
When radius is given:
12 in
Circumference
3 in
C = πd
C = 2πr
C = π(12 in)
C = 2π(3 in)
Exact answer:
Exact answer:
C = 12π in
C = 6π in
Approximate answer:
Approximate answer:
C ≈ (3.14)(12 in)
C ≈ 2(3.14)(3 in)
C ≈ 37.68 in
C ≈ 18.84 in
Validate:
Area
Validate:
Correct linear units: in 
Correct linear units: in 
Correct arithmetic:
Correct arithmetic:
37.68
= 3.14 
12
18.84
= 6.28
3
6.28
= 3.14 
2
A = πr2
A = πr2
r=½d
A = π(3 in)2
12
= 6 in
2
A = π(6 in)2
Exact answer:
r=
A = 9π in2
Approximate answer:
Exact answer:
A ≈ 9(3.14) in2
A = 36π in2
A ≈ 28.26 in2
Approximate answer:
Validate:
A ≈ 36(3.14) in2
Correct square units: in2 
A ≈ 113.04 in2
Correct arithmetic:
Validate:
Correct square units: in 
2
Correct arithmetic:
113.04
= 36 
3.14
28.26
=9 
3.14
Chapter 3 — Geometry
198
Models
Model 1
Find both the exact and approximate measurement of the circumference of a circle whose radius is 5 meters.
Exact answer
r=5m
C = 2πr
C = 10π m
Approximate
answer
Validate:
Compare answers using d
d = 2r = 2 × 5 m = 10 m
C = πd
C = π(10 m)
C = 10 π m 
C ≈ 10 (3.14) m
Correct linear units: m 
C ≈ 31.4 m
Correct arithmetic:
31.4
= 3.14 
10
Model 2
Find the exact and approximate area of a circle if the diameter is 42 inches.
Exact answer
r = (½)d
r = (½)(42) = 21
Compare answers using r
A = πr
d
A = π(r) = π  
2
2
A = π(21 in)2
A = 441π in2
Approximate
answer
Validate:
2
2
2
 42 
A = π 
 2 
 1764 
A = π

 4 
A = π441 or 441π 
A ≈ 441(3.14) in2
Correct square units: in2 
A ≈ 1384.74 in2
Correct arithmetic:
1384.74
= 441 
3.14
Section 3.3 — Circles
199
Model 3
If the circumference of a circle is 113.04 feet, what is its diameter?
C = rd
C
d=
r
113.04 ft
d.
3.14
d . 36 ft
Validate:
�d = C
3.14 × 36 = 113.04 
Addressing Common Errors
Issue
Using the
diameter
when the
radius is
needed
Incorrect
Process
Find the approximate Double-check the
area of a circle if the given information
against the
diameter is 7 cm.
A = πr
2
A ≈ 3.14 × 72 cm
A ≈ 153.86 cm2
Correct
Process
Resolution
formula.
Understanding
vocabulary is
key to correctly
working problems.
r = (½)d
r=
7
= 3.5 cm
2
A = π(r)
2
A . 3.14 × (3.5 cm) 2
Validation
Correct square
units:
cm2 
Correct arithmetic:
38.47
. 3.14 
12.25
A . 3.14 × (12.25 cm 2 )
A . 38.47 cm 2
Giving an
estimated
answer when
an exact
answer is
needed
Find the exact
circumference of a
circle whose radius
is 22 in.
C = 2πr
C = 2π(22 in)
C ≈ 2 × 3.14 × 22
C ≈ 138.16 in
Exact answers
for circumference
or area of a
circle will always
contain pi written
as π. The decimal
approximation for
π is not used for
an exact answer.
C = 2πr
C = 2π(22 in)
C = 44π in
Correct linear units:
in 
Correct arithmetic:
44r in
= 2r 
22 in
Chapter 3 — Geometry
200
Incorrect
Process
Issue
Giving an
exact answer
when an
estimated
answer is
needed
Resolution
Use the
Approximate the
approximate value
area of a circle
whose radius is 12 m. for π to calculate
A = πr2
an estimated
answer.
A= π × 12 m2
Find the area of a
circle with diameter
of 12 ft.
r = (1/2)d
r = 6 ft
A = πr2
A= π × 12 m2
A = 144π m2
A ≈ 144(3.14) m2
A ≈ 452.16 m2
A = 144π m2
Using the
wrong unit in
the answer
Correct
Process
An area requires
square units.
Multiply units
together as well
as constants and
variables.
r=½d
r = ½ (12)
r = 6 ft
A = π(6 ft)2
A= π(6 ft)(6 ft)
A = π6
2
Validation
Correct linear units:
m2 
Correct arithmetic:
452.16
= 144 
3.14
Correct linear units
ft2 
Correct arithmetic:
36
=6 
6
A= 36π ft2
A= 36π ft
Preparation Inventory
Before proceeding, you should be able to calculate or understand how to use the following:
Area of a circle
Circumference of a circle
Approximate value of pi (π)
Relationship between radius and diameter
The London Eye
Ferris Wheel
Constructed
in 1999, its
diameter is
135 meters.
Section 3.3
Activity
Circles
Performance Criteria
• Using the appropriate formula for circumference
– acurate calculation of an exact answer using �
– correct calculation of an approximate answer using 3.14 for �
– validation of the answer
• Calculating the area of a circle
– acurate calculations when given radius
– acurate calculations when given diameter
– acurate calculation of an exact answer using �
– correct calculation of an approximate answer using 3.14 for �
– validation of the answer
Critical Thinking Questions
1. Why is pi (�) an irrational number?
2. Why is pi (�) considered a constant?
3. What is the difference between an approximate answer and an exact answer?
201
202
Chapter 3 — Geometry
4. Why are linear units used for circumference?
5. Give five instances where it would be necessary to know the measurement of a circle’s circumference.
6. Why are square units used for the area of a circle?
7. Give five instances where it would be necessary to know the area of a circle.
8. Which measurement is most useful: diameter or radius?
Section 3.3 — Circles
Tips
for
203
Success
• Since half the diameter is equal to the radius and twice the radius is equal to the diameter, use the alternate
measurement and its corresponding formula for circumference and area when checking your arithmetic
Demonstrate Your Understanding
1. Find the circumference of a circle, given the following information:
Problem
a)
d = 3.2 meters
b)
d = 12 ½ inches
c)
r = 13.6 feet
d)
d=9
1
3
yards
Worked Solution
Validation
Chapter 3 — Geometry
204
2. Find the area of a circle, given the following information:
Problem
a)
d = 3.2 meters
b)
d = 12 ½ inches
c)
r = 13.6 feet
d)
d=9
1
3
Worked Solution
Validation
yards
3. Approximately how many yards of fringe are needed to finish the edge of a circular rug that is six feet in
diameter?
4. What is the radius of a surveyor’s wheel if its circumference is one yard? Give both exact and approximate
answers.
Section 3.3 — Circles
205
5. If a rotating sprinkler can spray water a distance of 12 feet, approximately how much area can be watered
in one revolution?
6. If the circumference of a tall pipe is 17.45 inches, what is its approximate diameter?
Identify
and
Correct
the
Errors
In the second column, identify the error(s) in the worked solution or validate its answer.
If the worked solution is incorrect, solve the problem correctly in the third column and validate your answer.
Worked Solution
1) Approximately how much
trim is needed to edge
a circular garden that
measures 7 feet across?
C = 7π
7π feet of trim
2) In the garden mentioned
above, how much black
plastic ground cover do
you need to completely
cover the garden?
A = (7)2 π
A ≈ 49 × 3.14 ft2
A ≈ 154 ft2
Identify Errors
or Validate
Correct Process
Validation
Chapter 3 — Geometry
206
Worked Solution
Identify Errors
or Validate
Correct Process
Validation
3) Find the exact
measurement of the area
of a circular swimming
pool that is 12 feet across.
r = ½ × 12 ft
r = 6 ft
A = πr2
A = π(6 ft)2
A = 36π ft2
4) Exactly how many square
feet of tarp are needed
to cover the pool in 3)
above?
Change π to 3.14:
A = 36π
A ≈ 36 × 3.14
113 feet of tarp
Graph of E8
E8 is a theoretical mathematical
structure that shows symmetrical
relationships in 248 dimensions.