Thank you for participating
in Teach It First!
This Teach It First Kit contains a Common Core Coach, Mathematics teacher lesson
followed by the corresponding student lesson. We are confident that using this
lesson will help you achieve your assessment preparation goals for your entire
class.
The Common Core Coach, Mathematics program is based on the philosophy
that mathematical skills are built on concepts. Math, maybe more than any other
school subject, builds from concept to concept, one on top of the other, over
several years. When students understand concepts and how they connect to
skills, they are better equipped to solve problems that they encounter in the real
world.
This program is 100% aligned to the Common Core State Standards and provides
a set of lessons for each of the five CCSS domains, with each lesson aligning
to one or more standard—together, lessons cover all the domain’s standards.
Concept Lessons begin with an underlying concept that connects directly to the
skill or skills taught in that lesson. Students will use a four-step problem-solving
process—Read, Plan, Solve, Check—to approach any mathematical problem.
Interactive questions follow examples and ask students to discuss a topic, model
a situation, try to solve a problem on their own, or check their work. With this
instructional anchor, you can implement the Common Core State Standards with
confidence.
We are happy to provide you this complimentary sample and would love to know
what you think. Once you have read through this lesson, do what you do best—
present it to your students. Then, don’t forget to complete a quick survey by
going to www.triumphlearning.com/CA/teach-it-first.
Regards,
Triumph Learning
Join the conversation about Common Core today by visiting commoncore.com, the
place where teachers, parents, and experts come together to share best practices and
practical information for successfully implementing Common Core standards in the
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21
Area and Circumference of Circles
Learning Objectives
• Students will derive the formula for finding the area of a circle by
dividing a circle into congruent wedges, rearranging them, and
using what they know about the circumference; students will use
formulas to calculate the circumference and area of a circle.
• Students will use formulas to find unknown lengths in circles and
use those lengths to find the circumference or area of a circle;
students solve real-world problems that involve finding the area
and circumference of circles.
Common Core State Standard
7.G.4 Know the formulas for
the area and circumference of
a circle and use them to solve
problems; give an informal
derivation of the relationship
between the circumference and
area of a circle.
Vocabulary
area the number of square units inside a figure
circumference the distance around a circle
Materials
• piece of string for each student (at least
12 centimeters long)
• ruler
• scissors (optional)
• Math Tool: Circumference and Area Formulas
(Circles)
• Math Tool: Spinners
Before the Lesson
Review the fact that the circumference of a circle is
the distance around it. Hand each student a piece
of string. Each student should turn to Math Tool:
Spinners and place the string so that it completely
covers the outside of one of the circular spinners.
Students can use their fingers to keep track of the
distance around the outside of the circle or they can
Understand
Connect
To help develop conceptual understanding of
the area of a circle, students can divide a circle
into smaller and smaller congruent wedges.
Step 1 shows dividing a circle into eight congruent
wedges. Be sure to point out to students that the
64
use scissors to cut the string so it is exactly as long
as the distance around one of the circles. Students
should then pull the string straight and measure its
length. Explain that this length, approximately
11 centimeters, is about equal to the circumference
of the circle.
Tell students that the diameter of the circular
spinner is 3.5 centimeters. Ask them to use the
formula for finding the circumference of a circle
and an approximation for p to estimate the
circumference of the circle: C 5 pd  3.14 3 3.5 
10.9 cm. Notice that students found the same, or
similar, estimates when they used a piece of string
as when they used the formula. Segue into the
lesson, explaining that now that students have
reviewed what the circumference of a circle is, they
can apply that knowledge to help them determine
a formula for finding the area of a circle.
distance around half of the circle is equal to half of
the circumference, or:
1
1
__
​ __
2  ​3 C 5 ​ 2  ​ (2pr) 5 pr units.
Students can use the diagram in steps 2 and 3 to
visualize how the wedges can be arranged to look
Duplicating any part of this book is prohibited by law.
radius the distance from the center of a circle to any point on
the circle
like a parallelogram and then rearranged to look
like a rectangle. Step 3 shows how the formula for
finding the area of a circle can be derived using the
formula for the area of a rectangle and the length
(pr units) and width (r units) of the rectangle.
To connect the concept to procedural
understanding, show how the formulas for finding
the circumference and area of a circle can be
applied to find the area of circle O. As you review
steps 2 and 3, explain that the  symbol means “is
approximately equal to.” Ask: Why is that symbol
Example
used in steps 2 and 3? Students should state that
whenever 3.14 is used for p, the measure being
calculated is only approximately equal to the true
measure. So, the  symbol is needed.
DISCUSS MP2 Students should state that the
formula C 5 pd could also be used to find the
circumference of circle O. The formula C 5 2pr
can be rewritten as C 5 2r ? p. Since d 5 2r, d can
be substituted for 2r. The formula then becomes
C 5 d ? p or C 5 pd.
CHECK
EXAMPLE 1
__
​ 2 ​ 3 154 5 77
The area of the smaller circle is less
than the area of the larger circle. Since
the area being subtracted from the
larger circle’s area is less than half its
area, I would expect my answer to be
greater than half the area of the larger
circle. My answer is greater than half
the area of the larger circle, so my
answer is reasonable. The area of the
placemat that is not covered by the
1
2
coaster is about 115 ​ __
2  ​in.
Problem Solving
Duplicating any part of this book is prohibited by law.
This is a multistep problem that applies the area
formulas to a real-world context.
PLAN
Then subtract the area of the smaller
(shaded) circle from the area of the
larger (white) circle.
SOLVE
The larger circle has r 5 7.
22
2
A  ​ ___
7   ​3 (7)
A  154 in.2
7
The smaller circle has d 5 7, so r 5 __
​ 2 ​.
As students are working, pay special attention
to problem 6. Because the diameter is a multiple
22
of 7, encourage students to use ___
​  7   ​for p. when
calculating this circumference.
1
2
A  38 ​ __
2 ​ in.
Students may use Math Tool: Circumference and
Area Formulas (Circles) for reference if they need to.
For answers, see page 112.
22
 7 
2
__
A  ​ ___
7   ​3 ​ ​ 2 ​  ​
Subtract to find the area of the placemat
that is not covered by the coaster.
1
1
2
__
154 2 38 ​ __
2  ​5 115 ​ 2  ​in.
Practice
65
Domain 4
This example differs from the previous
problem because it requires multiple steps.
Students must first substitute known values into the
formula for finding the circumference of a circle and
solve for r. Only when they know the value for r can
the approximate area of the circle be calculated.
TRY MP5 This problem differs from the problem
in the example because instead of giving an
approximate circumference, the circumference is
given in terms of p. Show how to calculate that the
radius is 14 units.
Work may vary. Possible work:
C 5 2pr
28p 5 2pr
28 5 2r
14 5 r
Does the smaller circle take up more
1
than or less than __
​ 2 ​ of the larger circle?
less than
If the fraction of the larger circle that
1
​ 2  ​, then I would
is shaded is less than __
expect that the area of the smaller
circle would be less than half of the
area of the larger circle.
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21 Area and Circumference of Circles
UNDERSTAND The circumference, C, is the distance around a circle. It can be found
using the formula C 5 2pr, where r is a radius of the circle and p is
22
an irrational number approximately equal to 3.14 or ​ ___
7   ​ .
The area of a circle is the total number of square units that fit inside the circle. Explain how you
can use what you know about the circumference of a circle to determine the formula for finding
its area.
1
Cut a circle into eight congruent wedges, as shown.
r
The distance from the center to any point on the circle measures r units.
The distance around the entire circle measures 2pr units. So, the
distance around half the circle is equal to half of that, or pr units.
2
r
Visualize reassembling the wedges to look like a parallelogram.
First, rearrange them to look like a parallelogram that has a length of approximately pr units. r
r
r
3
Divide one wedge in half. Move that half so the diagram resembles a rectangle.
r
r
r
If you continue to divide the circle into congruent wedges and the number of wedges
approaches infinity, the resulting figure would look more and more like a rectangle. Its length would be pr units and its width would be r units. So:
A of circle 5 length 3 width 5 pr ? r 5 pr2
▸ The area of a circle is found by the formula: A 5 pr2.
118 Domain 4: Geometry
Duplicating any part of this book is prohibited by law.
r
Connect
Find the approximate circumference and area of circle O.
10
1
cm
O
Determine the length of the radius of circle O.
The line segment labeled 10 cm has
endpoints on the circle and passes
through the center, point O. The line
segment is a diameter of the circle.
The radius of a circle is equal to half its
diameter, so:
r 5 10 4 2 5 5 cm
2
Use the formula C 5 2pr to find
the approximate circumference. Use 3.14 for p.
C 5 2pr
C < 2 3 3.14 3 5
C < 31.4 cm
3
Use the formula A 5 pr 2 to find the
approximate area.
Note: You are using an approximation
for p, so the circumference you found,
31.4 cm, is also an approximation.
A 5 pr 2
A < 3.14 3 52
A < 3.14 3 25
Duplicating any part of this book is prohibited by law.
A < 78.5 cm2
Again, 78.5 cm2 is only an approximation
of the area because you used an
approximate value for p.
▸ The approximate circumference
is 31.4 centimeters, and the
approximate area is 78.5 square centimeters.
CU S S
DIS
Could you also use the formula C 5 pd
to find the circumference of circle O?
Explain.
Lesson 21: Area and Circumference of Circles 119
EXAMPLE The circumference of a circle is approximately 50.24 meters. What is its
approximate area?
1
Substitute known values into the
equation for the circumference of a circle.
Substitute 50.24 for C and 3.14 for p.
C 5 2pr
2
Solve for r.
50.24 < 2 3 3.14 3 r
50.24 < 2 3 3.14 3 r
50.24 < 6.28r
50.24 _____
6.28r
_____
​  6.28   
​ < ​ 6.28  ​
8<r
3
Substitute that value for r in the area
formula.
A 5 pr 2
A < 3.14 3 82
A < 200.96
▸ The approximate area of the circle is
200.96 square meters.
The circumference of a circle is 28p inches. What is its radius?
Show your work.
120 Domain 4: Geometry
Duplicating any part of this book is prohibited by law.
TRY
Problem Solving
read
A circular coaster, with a diameter of 7 inches, is positioned on a white placemat. The placemat is a circle whose radius is equal to the diameter of the coaster. About how many square inches of the placemat are not covered by the coaster?
7 in.
plan
Find the area of both circles.
Then subtract the area of the
circle from the area of the
circle.
solve
22
Find the approximate area of each circle. Use ​ ___
7   ​ for p since the radius is a multiple of 7.
The larger circle has r 5
.
The smaller circle has d 5
A of larger circle 5 pr 2
22
A < ​ ___
7   ​ 3 ( .
A of smaller circle 5 pr 2
22
A < ​ ___
7   ​ 3 ( )2
in.2
A<
, so r 5
A<
)2
in.2
Subtract to find the area of the placemat that is not covered by the coaster.
2
5
in.2
check
Look at the diagram.
1
Does the smaller circle take up more than or less than ​ __
2 ​  of the larger circle?
Duplicating any part of this book is prohibited by law.
1
If the fraction of the larger circle that is shaded is
than ​ __
2 ​  , then I would expect
that the area of the smaller circle would be
than half of the area of the larger
circle.
1
​ __
2 ​  3
5
▸ The area of the smaller circle is
than the area of the larger circle. Since the
area being subtracted from the larger circle‘s area is
than half its area,
I would expect my answer to be
than half the area of the larger circle. My answer is
than half the area of the larger circle, so my answer
reasonable. The area of the placemat that is not covered by the coaster
is about
in.2.
Lesson 21: Area and Circumference of Circles 121
Practice
A radius, r, or diameter, d, of a circle is given. Find the other measure.
1.
If d 5 4, r 5
.
If r 5 4, d 5
2.
.
3.
If d 5 3, r 5
.
Find the approximate circumference of each circle shown. Show your work.
4.
5.
6.
28 in.
6 ft
H
IN
T
10 in.
22  ​ as an
Use 3.14 or ​ __
7
approximation for p.
Find the approximate area of each circle shown. Show your work.
2 yd
8.
9.
9m
20 ft
Fill in the blank with an appropriate word or phrase.
10. The radius of a circle is equal to
11. The circumference of a circle is the
12. The area of a circle is the number of
122 Domain 4: Geometry
its diameter.
around it.
that fit inside it.
Duplicating any part of this book is prohibited by law.
7.
Choose the best answer.
13. Circle A has a circumference of
36p meters. What is the radius
of circle A?
14. Circle O has a circumference of
132 feet. What is the approximate area of circle O?
A.   2 m
B.   6 m
A. 5,544 ft
2
B. 1,386 ft2
C. 12 m
D. 18 m
C.    441 ft2
D.   346.5 ft2
Solve.
15. Jordan has a circular flower bed that
is 18 feet in diameter in her yard. Approximately how many square feet of
her yard is covered by the flower bed?
16. A circular fountain in a park is 30 feet
in diameter. A park employee will plant marigolds around it. What is the circumference of the fountain?
Flower
bed
30 ft
18 ft
Duplicating any part of this book is prohibited by law.
17.
Show Two flat, circular plates are
placed on a circular tabletop. The
diameter of each plate equals the radius
of the tabletop. How many square
inches of the tabletop are not covered
by the plates? What fraction of the
tabletop is covered? Show your work.
18.
EXPLAIN This dartboard is divided into
three regions. The outermost region
has dark shading, the middle region is lightly shaded, and the center region
is unshaded. What is the area of the
region with the dark shading? Explain
how you determined your answer.
1 in.
1 in.
12 inches
1 in.
Lesson 21: Area and Circumference of Circles 123