6-4 : Learn to find the area and circumference of circles. Area and Circumference of Circles (including word problems)

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Area and Circumference of Circles (including word problems)

6-4 : Learn

to find the area and circumference of circles.

Circles

8-3 Learn

to find the Circumference of a circle.

8-6 Learn

to find the area of circles.

Radius

Center

Diameter

Circumference

The diameter d is twice the radius r.

d = 2 r

The circumference of a circle is the distance around the circle.

C d The distance around a circle is called circumference. For every circle, the ratio of circumference C to diameter d is the same. This ratio, , is represented by the symbol π, called pi. Pi is approximately equal to 3.14 or²². By multiplying both sides of the 7 equation ^C

d= π by d, you get the formula for circumference, C = πd, or C = 2πr.

Radius

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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Additional Example 1: Finding the Circumference of a Circle

A. Circle with a radius of 4 m C = 2πr

= 2π(4)

= 8π m

B. Circle with a diameter of 3.3 ft C = πd

= π(3.3)

= 3.3π ft

Find the circumference of each circle in terms of ππππ.

Try This: Example 1

A. Circle with a radius of 8 cm C = 2πr

= 2π(8)

= 16π cm

B. Circle with a diameter of 4.25 in.

C = πd

= π(4.25)

= 4.25π in

Find the circumference of each circle in terms of ππππ.

Find the circumference of the circle to the nearest tenth. Use 3.14 for

π π π π

.

Additional Example 3A: Finding the Circumference of a Circle

12 in.

C =

π

_d C ^≈3.14 · 12 C ^≈37.68

You know the diameter.

Substitute for π and d.

Multiply.

The circumference of the circle is about 37.7 in.

A.

π π π π

.

Additional Example 3B: Finding the Circumference of a Circle

18 cm C = 2πr C ≈ 2 · 3.14 · 18 C ≈ 113.04

You know the radius.

Substitute for πand r.

Multiply.

The circumference of the circle is about 113.0 cm.

B.

π π π π

.

Try This: Example 3A

7 in.

C =

π

_d C ^≈3.14 · 7 C ^≈21.98

You know the diameter.

Substitute for π and d.

Multiply.

The circumference of the circle is about 22 in.

A.

π π π π

.

Try This: Example 3B

11 cm C = 2πr C ≈ 2 · 3.14 · 11 C ≈ 69.08

You know the radius.

Substitute for πand r.

Multiply.

The circumference of the circle is about 69.1 cm.

B.

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Additional Example 2: Finding the Area of a Circle

A = πr²= π(4²)

= 16π in²

A. Circle with a radius of 4 in.

Find the area of each circle in terms of π.π.π.π.

B. Circle with a diameter of 3.3 m A = πr²= π(1.65²)

= 2.7225π m²

d 2 = 1.65

B. Circle with a diameter of 2.2 ft A = πr²= π(1.1²)

= 1.21π ft²

d 2 = 1.1 Try This: Example 2 Find the area of each circle in terms of π.π.π.π.

A = πr²= π(8²)

= 64π cm²

A. Circle with a radius of 8 cm

A circle can be cut into equal- sized 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 one-

half the circumference C of the So the area of the parallelogram can be written as

A = bh, or A = ¹ 2Cr.

Since C = 2

π

_{r, A =}¹

2(2

π

_{r)r =}

π

r². circle.

AREA OF A CIRCLE The area A of a

circle is the product of

π

_and the square of the circle’s radius r.

•

^r

A =

π

_r²

Find the area of the circle to the nearest tenth.

Use 3.14 for

π

.

Additional Example 1A: Finding the Area of a Circle

A.

7 cm

A =

π

_r² A ≈ 3.14 · 7² A ≈ 3.14 · 49 A ≈ 153.86

The area of the circle is about 153.9 cm^2.

Use the formula.

Substitute 7 for r.

Evaluate the power.

Multiply.

The order of operations calls for evaluating the exponents before multiplying.

Remember!

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Use 3.14 for

π π π π

.

Additional Example 1B: Finding the Area of a Circle

B.

18 ft

A =

π

_r² A ≈ 3.14 · 9² A ≈ 3.14 · 81 A ≈ 254.34

The area of the circle is about 254.3 ft^2.

Use the formula.

Multiply.

Use 3.14 for

π π π π

. A.

10 cm

A =

π

_r² A ≈ 3.14 · 10² A ≈ 3.14 · 100 A ≈ 314

The area of the circle is about 314 cm^2.

Use the formula.

Multiply.

Try This: Example 1A

Use 3.14 for

π π π π

. B.

12 ft

A =

π

_r² A ≈ 3.14 · 6² A ≈ 3.14 · 36 A ≈ 113.04

The area of the circle is about 113.0 ft^2.

Use the formula.

Multiply.

Try This: Example 1B

The diameter of a circular pond is 42 m. What is its circumference? Use

Additional Example 4: Application

22 7

for

π π π π

_.

C = πd You know the diameter.

C ≈ 22

7 · 42 Substitute ²²

7for π and 42 for d.

C ≈ 22 7· 42

1 C ≈ 22

7· 42 1

6 1 C ≈ 132

The circumference of the pond is about 132 m.

Write 42 as a fraction.

Simplify.

Multiply.

The diameter of a circular spa is 14 m. What is its circumference? Use

Try This: Example 4

22 7

for

π π π π

_.

C = πd You know the diameter.

C ≈ 22

7 · 14 Substitute ²²₇for π and 14 for d.

C ≈ 22 7· 14

1 C ≈ 22

7· 14 1

2 1 C ≈ 44

The circumference of the spa is about 44 m.

Write 14 as a fraction.

Simplify.

Multiply.

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 7for

π π π π

_. A =

π

r²

A ≈ 22 7 ·14² A ≈ 22

7 ·196 A ≈ 22 · 28 A ≈ 616

The area of the top that will cover the drain is about 616 in². 1

28

Use the formula for the area of a circle.

Substitute. Use 14 for r.

Multiply.

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22 7 for ππππ. A = π r²

A ≈ 22 7 ·7² A ≈ 22

7 · 49 A ≈ 22 · 7 A ≈ 154

1 7

Multiply.

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

The area of the top that will cover spa is about 154 ft².

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.

A = π r² A =

π

_{· 25}² A ≈ 1963.495408 A ≈ 1,963

Use a calculator. ^π × 25 ^x² Round.

The sprinkler covers about 1,963 ft².

A = π r² A =

π

_{· 13}² A ≈ 530.9216 A ≈ 531

Use a calculator. ^π × 13 ^x² Round.

The pond has a surface area of about 531 ft². Try This: Example 3

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.

Additional Example 4: Measurement Application

C = πd = π(56)

≈176 ft

≈ ²²₇ (56) ^≈

22 7 A Ferris wheel has a diameter of 56 feet and makes 15 revolutions per ride. How far would someone travel during a ride? Use for ππππ.

Find the circumference.

56 1 22

7

The distance is the circumference of the wheel times the number of revolutions, or about 176 • 15 = 2640 ft.

12

3

6 9

Try This: Example 4

A second hand on a clock is 7 in long. What is the distance it travels in one hour? Use for ππππ.

22 7

C = πd = π(14)

≈ ²²₇ (14) ^≈

Find the circumference.

≈44 in.

The distance is the circumference of the clock times the number of revolutions, or about 44 • 60 = 2640 in.

14 1 22

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