Circles
6-4 : Learn to find the area and
circumference of circles.
Area and Circumference of Circles
(including word problems)
8-3 Learn to find the Circumference of a
circle.
8-6 Learn to find the area of circles.
Radius
Center
Diameter
The diameter d is
twice the radius r.
d = 2r
Circumference
The distance around a circle is
called circumference. For
every circle, the ratio of
circumference C to diameter d
is the same. This ratio, C , is
represented by the symbol π,
called pi. Pi is approximately
equal to 3.14 or 22 . By
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multiplying both sides of the
equation C = π by d, you get
d
The circumference of a circle is the distance
around the circle.
Radius
d
the formula for circumference,
C = πd, or C = 2πr.
Diameter
Circumference
Remember!
Pi (π) is an irrational number that is often
approximated by the rational numbers 3.14
and 22 .
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1
Additional Example 1: Finding the Circumference of a
Circle
Find the circumference of each circle in
terms of π.
Try This: Example 1
Find the circumference of each circle in
terms of π.
A. Circle with a radius of 4 m
C = 2πr
= 2π(4)
= 8π m
A. Circle with a radius of 8 cm
C = 2πr
= 2π(8)
B. Circle with a diameter of 3.3 ft
C = πd
= π(3.3)
= 3.3π ft
B. Circle with a diameter of 4.25 in.
C = πd
= π(4.25)
= 4.25π in
= 16π cm
Additional Example 3A: Finding the Circumference
of a Circle
Find the circumference of the circle to the
nearest tenth. Use 3.14 for π.
A.
12 in.
C = πd
You know the diameter.
C ≈ 3.14 · 12 Substitute for π and d.
C ≈ 37.68
Additional Example 3B: Finding the Circumference
of a Circle
Find the circumference of the circle to the
nearest tenth. Use 3.14 for π.
B.
18 cm
Multiply.
The circumference of the circle is about 37.7 in.
7 in.
C = πd
C ≈ 3.14 · 7
You know the diameter.
C ≈ 21.98
Multiply.
Substitute for π and d.
The circumference of the circle is about 22 in.
You know the radius.
C ≈ 2 · 3.14 · 18 Substitute for π and r.
C ≈ 113.04
Multiply.
The circumference of the circle is about 113.0 cm.
Try This: Example 3A
Find the circumference of the circle to the
nearest tenth. Use 3.14 for π.
A.
C = 2πr
Try This: Example 3B
Find the circumference of the circle to the
nearest tenth. Use 3.14 for π.
B.
11 cm
C = 2πr
You know the radius.
C ≈ 2 · 3.14 · 11 Substitute for π and r.
C ≈ 69.08
Multiply.
The circumference of the circle is about 69.1 cm.
2
Additional Example 2: Finding the Area of a Circle
Find the area of each circle in terms of π.
A. Circle with a radius of 4 in.
A = πr2 = π(42)
= 16π in2
B. Circle with a diameter of 3.3 m
d
= 1.65
2
A = πr2 = π(1.652)
= 2.7225π m2
Try This: Example 2
Find the area of each circle in terms of π.
A. Circle with a radius of 8 cm
A = πr2 = π(82)
= 64π cm2
B. Circle with a diameter of 2.2 ft
d
2
= 1.1
A = πr2 = π(1.12)
A circle can be cut into equalsized sectors and arranged to
resemble a parallelogram. The
height h of the parallelogram is
equal to the radius r of the
circle, and the base b of the
parallelogram is equal to onehalf the circumference C of the circle. So the area of
the parallelogram can be written as
A = bh, or A = 1 Cr.
2
= 1.21π ft2
Since C = 2πr, A = 1 (2πr)r = πr2.
2
Additional Example 1A: Finding the Area of a Circle
Find the area of the circle to the nearest tenth.
Use 3.14 for π.
AREA OF A CIRCLE
The area A of a
circle is the
product of π and
the square of the
circle’s radius r.
A = πr2
A.
A = πr2
•
r
7 cm
Use the formula.
A ≈ 3.14 · 72
Substitute 7 for r.
Evaluate the power.
Multiply.
A ≈ 153.86
The area of the circle is about 153.9 cm2.
A ≈ 3.14 · 49
Remember!
The order of operations calls for evaluating the
exponents before multiplying.
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Additional Example 1B: Finding the Area of a Circle
Try This: Example 1A
Find the area of the circle to the nearest tenth.
Use 3.14 for π.
Find the area of the circle to the nearest tenth.
Use 3.14 for π.
A = πr2
B.
18 ft
A ≈ 3.14 ·
Use the formula.
92
A ≈ 3.14 · 81
Substitute 9 for r.
Evaluate the power.
Multiply.
A ≈ 254.34
The area of the circle is about 254.3 ft2.
A = πr2
A.
10 cm
Substitute 10 for r.
Evaluate the power.
Multiply.
A ≈ 314
The area of the circle is about 314 cm2.
A ≈ 3.14 · 100
Additional Example 4: Application
Try This: Example 1B
Find the area of the circle to the nearest tenth.
Use 3.14 for π.
B.
12 ft
A = πr2
Use the formula.
A ≈ 3.14 · 62
Substitute 6 for r.
Evaluate the power.
Multiply.
A ≈ 3.14 · 36
A ≈ 113.04
The area of the circle is about 113.0 ft2.
Use the formula.
A ≈ 3.14 · 102
The diameter of a circular pond is 42 m. What
is its circumference? Use 22 for π.
7
C = πd
You know the diameter.
C ≈ 22 · 42
7
C ≈ 22 · 42
7 1
22 · 42 6
C≈ 7 1
1
C ≈ 132
Substitute 22 for π and 42 for d.
7
Write 42 as a fraction.
Simplify.
Multiply.
The circumference of the pond is about 132 m.
Try This: Example 4
The diameter of a circular spa is 14 m. What
is its circumference? Use 22 for π.
7
C = πd
You know the diameter.
C ≈ 22 · 14
7
C ≈ 22 · 14
7 1
22 14 2
C≈ 7 · 1
1
C ≈ 44
Substitute 22 for π and 14 for d.
7
Write 14 as a fraction.
Simplify.
Multiply.
The circumference of the spa is about 44 m.
Additional Example 2: Application
Park employees are fitting a top over a
circular drain in the park. If the radius of the
drain is 14 inches, what is the area of the top
that will cover the drain? Use 22 for π.
7
A = π r2
Use the formula for the area of a circle.
Substitute. Use 14 for r.
A ≈ 22 ·142
7
28
A ≈ 22 ·196
Evaluate the power.
17
A ≈ 22 · 28
A ≈ 616
Multiply.
The area of the top that will cover the drain is about 616 in2.
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Additional Example 3: Application
Try This: Example 2
Albert was designing a cover for a spa. If the
radius of the spa is 7 ft, what is the area of
the cover that will be made? Use 22 for π.
7
A = π r2
A ≈ 22 ·72
7
7
A ≈ 22 · 49
17
A ≈ 22 · 7
Use the formula for the area of a circle.
Substitute. Use 7 for r.
A = π r2
Evaluate the power.
A=
Multiply.
A ≈ 154
A golf course is irrigated with sprinklers that
spray in a circle. If a sprinkler waters a
circular area with a radius of 25 feet, how
many square feet does the sprinkler cover?
Round your answer to the nearest whole
number.
The area of the top that will cover spa is about 154 ft2.
Try This: Example 3
Use the formula for the area of a circle.
π · 252
Substitute. Use 25 for r.
A ≈ 1963.495408
Use a calculator. π
A ≈ 1,963
Round.
×
25 x2
The sprinkler covers about 1,963 ft2.
Additional Example 4: Measurement Application
Janet had a circular fish pond in her back
yard. She wanted to find the surface area of
the pond. If it had a radius of 13 feet, what is
its surface area in square feet? Round your
answer to the nearest whole number.
A Ferris wheel has a diameter of 56 feet
and makes 15 revolutions per ride. How far
would someone travel during a ride? Use 22
7
for π.
A = π r2
C = πd = π(56)
A=
Use the formula for the area of a circle.
π · 132
Substitute. Use 13 for r.
A ≈ 530.9216
Use a calculator. π
A ≈ 531
Round.
×
13 x2
The pond has a surface area of about 531 ft2.
Find the circumference.
22
22
≈
(56) ≈ 7 56 ≈ 176 ft
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1
The distance is the circumference of the
wheel times the number of revolutions, or
about 176 • 15 = 2640 ft.
Try This: Example 4
A second hand on a clock is 7 in long. What
is the distance it travels in one hour? Use 22
7
for π.
C = πd = π(14)
12
9
≈
3
6
Find the circumference.
22
22
(14) ≈ 7 14
7
1
≈ 44 in.
The distance is the circumference of the
clock times the number of revolutions, or
about 44 • 60 = 2640 in.
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