11.1
Radius, Diameter, and
Circumference of a Circle
Lesson Objectives
Le
•
•
•
•
Identify parts of a circle.
Recognize that a circle’s diameter is twice its radius.
Use formulas to find the circumference of a circle.
Identify semicircles and quarter circles, and find the distance
around them.
Vocabulary
center
diameter
arc
quadrant
radius, radii
circumference
semicircle
arn Identify the center and radius of a circle.
These are circles.
You can use a compass to draw a circle. Notice that all points on the circle are the
same distance from the center.
O
Q
P
R
O is the center of the circle.
OP is a radius of the circle.
OQ and OR are also radii of the circle.
The plural of radius is radii.
In a given circle, all radii
have the same length.
OP 5 OQ 5 OR
A radius is a line segment connecting the
center and a point on the circle.
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Le
arn Identify the diameter of a circle.
In the circle below, O is the center. The line segments PQ and RS pass through the
center O. PQ is a diameter of the circle. RS is another diameter of the circle. TU is
not a diameter.
A diameter of a circle is a line segment
that connects two points on the circle
and passes through its center.
S
P
U
O
Q
All diameters of a given circle
have the same length.
R
T
OP , OQ , OR , and OS are radii of the circle.
OP 5 OQ 5 OR 5 OS
Math Note
Diameter 5 2 · radius
Radius 5 Diameter 4 2
So PQ5 2 · OP
5 2 · OQ
5 2 · OR
5 2 · OS
Guided Practice
Complete.
A
1 In the figure, O is the center of the circle with AB , CD , and ED as shown. a) Name all the diameters that are drawn in the circle.
b) Which line segment that joins two points on the circle is not a diameter?
Explain why it is not a diameter.
2 The radius of a circle is 6 centimeters. What is the length of its diameter?
C
O
D
E
B
Diameter
5 2 · radius
5
?
?
5
?
cm
?
The diameter of the circle is
?
centimeters.
3 The diameter of a circle is 15 inches. What is the length of its radius?
Radius
5 diameter 4 2
5
?
?
5
?
in.
?
The radius of the circle is
?
inches.
Lesson 11.1 Radius, Diameter, and Circumference of a Circle
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Material:
• compass
DRAWING CIRCLES USING A COMPASS
ST E P
1 Measure 5 centimeters on a ruler with a compass.
Example
0
cm
1
2
3
4
5
6
7
8
9
10
11
12
13
14
ST E P
2 Then draw a circle. Draw and label the center O and four radii:
OP , OQ , OR , and OS .
Example
ST E P
3 Measure the radii OP , OQ , OR , and OS . What can you
say about the lengths of OP , OQ , OR , and OS ?
Q
O
Le
R
P
S
arn Identify the circumference of a circle.
The circumference of a circle is
the distance around the circle.
124
The hoop in the photo has the shape of a circle.
The distance around the hoop is called its circumference.
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INVESTIGATING THE RELATIONSHIP BETWEEN THE CIRCUMFERENCE AND
DIAMETER OF A CIRCLE
ST E P
1 Lisa uses a string to measure the circumference of each circle to the nearest
tenth of a centimeter and records it in a table.
2 cm
3 cm
A
B
4 cm
5 cm
C
D
Copy the table. Divide the circumference of each circle by its diameter.
Round your answers to the nearest tenth. Record your results.
Circle
Diameter (cm)
Circumference (cm)
Circumference 4 diameter
A
2
6.2
?
B
3
9.4
?
C
4
12.5
?
D
5
15.6
?
What do you notice about the quotients in the last column?
The circumference of any circle divided by its diameter always
gives the same value.
The Greek letter p is used to
The letter p is
represent this value.
pronounced “pie.”
Continue on next page
Lesson 11.1 Radius, Diameter, and Circumference of a Circle
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ST E P
2 To see the value of p up to 9 decimal places,
press
ENTER
on your calculator. Round the value of p to
a) the nearest tenth.
b) the nearest hundredth.
c)
the nearest thousandth.
Any value of p is an
approximation. The value
of p is often approximated
as 3.14 or 22 .
7
ST E P
ST E P
3 In 1 , you learned that the circumference of any circle
divided by its diameter is equal to p. You can use this
fact to write a formula for the circumference of a circle.
Complete the following statement.
Since circumference 4 diameter 5 p,
Circumference 5 p ·
You also know that the diameter of a circle is 2 times its radius. You can
use this fact to write a related formula for the circumference of a circle.
Complete the following statement.
?
Circumference
5 p · diameter
5p·2·
?
52·p·
?
Using C for circumference, d for diameter, and r for radius, you can write
these formulas as
Math Note
C 5 pd
C 5 2pr
126
pd means p · d and
2pr means 2 · p · r.
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Le
arn Find the circumference of a circle.
a) Find the circumference of the plate shown. Use
22
as an approximation for p.
7
28 cm
Math Note
Circumference 5 pd The symbol  means
“approximately equal to.”
Write formula.

22
· 28
7
5
28
22
·
1
7
1
Substitute.
4
Divide by the common factor, 7.
5 22 · 4 5 88 cm
Simplify.
Multiply.
The circumference of the plate is approximately 88 centimeters.
b)
The radius of a bicycle wheel is 10.5 inches. Find the circumference of the wheel.
Use 3.14 as an approximation for p.
Circumference
5 2pr  2 · 3.14 · 10.5 5 65.94 in.
Write formula.
Substitute.
Multiply.
The circumference of the bicycle wheel is approximately 65.94 inches.
Guided Practice
Copy and complete the table. Use
4
22
as an approximation for p.
7
Circle
Radius (cm)
Diameter (cm)
Circumference (cm)
A
?
14
?
B
21
?
?
C
10.5
?
?
Lesson 11.1 Radius, Diameter, and Circumference of a Circle
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Copy and complete the table. Use 3.14 as an approximation for p.
Le
5
Circle
Radius (cm)
Diameter (cm)
Circumference (cm)
D
?
25
?
E
16
?
?
F
8.25
?
?
arn Recognize that half of a circle is a semicircle and
a quarter of a circle is a quadrant.
You can divide a circle into halves. Each half circle is called a semicircle.
B
A
B
A
C
C
ABC is an arc of
the circle. It is a
semicircular arc.
The length of the semicircular arc ABC is half the circumference
of the circle.
You can divide a circle into quarters. Each quarter circle is
called a quadrant.
Q
R
Q
R
PQR is also an arc.
It is the arc of
a quadrant.
P
P
The length of arc PQR of the quadrant is one quarter
the circumference of the circle.
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Le
arn Find the lengths of a semicircular arc and the arc of a quadrant.
a)
A length of wire is bent into a semicircular arc. The length of EF is 21 centimeters.
Find the length of the wire. Use
22
as an approximation for p.
7
21 cm
E
F
Circumference
5 pd 22
· 21 7

5 66 cm
Length of semicircular arc5
Write formula.
Substitute.
Multiply.
1
3 66
2
5 33 cm
Length of wire5 Length of semicircular arc
5 33 cm
The length of the wire is approximately 33 centimeters.
b)
A circular ring of radius 5 inches is cut into four equal parts. Find the length of
each arc of a quadrant. Use 3.14 as an approximation for p.
5 in.
Circumference
5 2pr  2 · 3.14 · 5
5 31.4 in.
Write formula.
Substitute.
Multiply.
1
3 31.4
4
Length of each arc of the quadrant5
5 7.85 in.
The length of each arc of a quadrant is approximately 7.85 inches.
Lesson 11.1 Radius, Diameter, and Circumference of a Circle
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Guided Practice
Complete.
6 A circular hoop is cut into two equal parts. Its diameter is 35 inches. Find the
length of each semicircular arc. Use
Circumference of hoop5 pd

?
?
5
?
in.
Length of each semicircular arc 5
22
as an approximation for p.
7
?
1
· circumference of hoop
2
5
?
?
5
?
in.
?
The length of each semicircular arc is approximately
?
inches.
7 A quadrant is cut from a square. The side of the square is 10 centimeters.
Find the length of the arc of the quadrant. Use 3.14 as an approximation
for p.
10 cm
Circumference of circle5 2pr
2·
5
?
?
·
?
cm
Length of the arc of the quadrant 5 circumference 4 4
5
?
4
5
?
cm
130
?
The length of the arc of the quadrant is approximately
?
centimeters.
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Materials:
•
•
•
•
DRAWING SEMICIRCLES AND QUADRANTS
ST E P
1 Use a compass to draw a circle of radius 2 inches.
compass
drawing triangles
ruler
protractor
Label the center O.
ST E P
2 Draw and label a diameter of the circle PQ. What do you notice?
?
A diameter of a circle divides it into
This figure is one of the semicircles.
Find the distance around the semicircle.
Use 3.14 as an approximation for p.
semicircles.
P
Q
O
2 in.
ST E P
3 In your circle, draw a second diameter perpendicular to PQ using a ruler and
protractor, or a drawing triangle. Label it RS . What do you notice?
a) Two perpendicular diameters of a circle divide it into
This figure is one of the quadrants.
Find the distance around the quadrant.
Use 3.14 as an approximation for p.
?
quadrants.
R
2 in.
Q
O
b)
This figure is made up of the semicircular arc,
the arc of a quadrant, and the radii OR and OQ .
Find the distance around the figure. Use 3.14 as
an approximation for p.
R
2 in.
O
Q
S
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Practice 11.1
Use the figure to complete. In the figure, O is the center of the circle and XY
is a straight line.
X
Z
O
Y
1 OX , OY , and OZ are ? of the circle.
2 XY is a ? of the circle.
3 OX 5 ? 5 ?
4 XY 5 ? · OZ
5 Circumference of the circle 5 p · ?
Find the circumference of each circle. Use
6 22
as an approximation for p.
7
7 21 in.
7 cm
8 9 2.8 ft
49 m
132
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Find the length of each arc. Use
22
as an approximation for p.
7
10 11 7.7 cm
9.8 ft
12 13 11.2 cm
2.1 m
Find the distance around each semicircle. Use 3.14 as an approximation for p.
14 15 20 in.
10 cm
16 17 25 m
7.5 ft
Find the distance around each quadrant. Use
18 3.5 in.
22
as an approximation for p.
7
19 10.5 cm
20 21 17.5 m
28 ft
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Solve. Show your work. Use 3.14 as an approximation for p.
22 A circular garden has a diameter of 120 feet. Find its circumference.
23 A circular coaster has a radius of 4 centimeters. Find its circumference.
24 The diameter of a roll of tape is 5
1
centimeters. Find its circumference.
2
25 The shape of a floor mat is a semicircle. Find the distance around the mat.
38 cm
38 cm
26 A small playground is shaped like a quadrant, as shown. Find the distance
around the playground.
P
24 ft
O
Q
24 ft
27 If the radius of a wheel is 14 inches, what is the distance traveled when the
wheel turns around 100 times?
14 in.
Find the distance around each figure. Use
22
as an approximation for p.
7
28 The figure is made up of a rectangle and a semicircle.
28 cm
12 cm
134
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29 The figure is made up of an equilateral triangle and a semicircle.
42 cm
Find the distance around each figure. Use 3.14 as an approximation for p.
30 The figure is made up of a rectangle and two identical semicircles.
32 ft
54 ft
31 The figure is made up of two identical quadrants, a semicircle, and an
equilateral triangle.
23 ft
23 ft
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