circumference and arc length

MM2G3 Students will understand properties of circles.

MM2G3 c Use the properties of circles to solve problems involving the length of an arc and the area of a sector.

Circumference and Arc Length

Lesson 6.7

Friday, April 21, 2023

How do we find the circumference of a circle and the length of an arc of a circle?

M2 Unit 3: Day 9



1. Find w, y, and z. Give the answers in simplest radical form.

Warm Ups



Warm Up

Find each measure.

54 2. RT

3. UQ

Warm Ups



Circumference The distance around a circle is called the circumference. The distance across a circle through the center is called the diameter. Pi is the ratio of the circumference of a circle to the diameter. Thus, for any circle, if you divide the circumference by the diameter, you get a value close to Pi.

C

d



The key gives the best possible approximation for on your calculator.Always wait until the last step to round.

Helpful Hint



EXAMPLE Use the formula for circumference

Find the indicated measures.

Write circumference formula.

Substitute 9 for r.

Simplify.

Use a calculator.

= 2 π 9

= 18

56.55

a. Circumference of a circle with radius 9 centimeters

SOLUTION

The circumference is about 56.55 centimeters.

ANSWER

a. C = 2 πr



Use the formula for circumference


Substitute 26 for c.

Divide each side by 2π.

Use a calculator.r4.14

= 2 πr26

=262π

r

C = 2 πr

The radius is about 4.14 meters.

ANSWER

Find the indicated measures.

b. Radius of a circle with circumference 26 meters

EXAMPLE



GUIDED PRACTICE

1. Find the circumference of a circle with diameter 5 inches. Find the diameter of a circle with circumference 17 feet.


Substitute 5 for d.

Use a calculator.

C = d

= 5

15.71The circumference is about 15.71 in.

SOLUTION

Guided Practice



GUIDED PRACTICE

Write circumference formula.C = d

circumference = 17 feet

Substitute 17 for c.

Divide each side by π.

Use a calculator.

The Diameter is about 5.41 feet.

r5.41

= d17

=17 d

1. Find the circumference of a circle with diameter 5 inches. Find the diameter of a circle with circumference 17 feet.



The arc length of a circle is a fraction of the circumference of the circle.



EXAMPLE 3Find arc lengths

Find the length of each red arc.

a.

SOLUTION

a. Arc length of AB 2π(8)60°

360°= ≈ 8.38 centimeters

EXAMPLE





b.

SOLUTION

b. Arc length of EF 2π(11)60°


EXAMPLE





c.

SOLUTION

c. Arc length of GH 2π(1)120°


EXAMPLE



EXAMPLE 4

Find arc lengths to find measuresFind the indicated measure.

d. Circumference C of ZSOLUTION

37.71 = C

4.19C 19

1=

Arc length of XYC

d.360°m XY=

4.19C 360°

40°=

EXAMPLE



EXAMPLE 4

Find arc lengths to find measures

Find the indicated measure.

e. m RS

442 (15.28)

360° = m RS

165° m RS

Arc length of RS2 r 360°

e. m RS=

442 (15.28)

m RS360°

=

SOLUTION

EXAMPLE



GUIDED PRACTICE

Find the indicated measure.

2. Length of PQ

Arc length of PQ =360°

(9) 5.89 yd75°

Guided Practice



GUIDED PRACTICE

3. Circumference of N

Arc length of LM = m LMC 360°

61.26 = 270°

C 360°

61.26 = 27C 36

81.68 m = C

Guided Practice



GUIDED PRACTICE

Arc length of EF = m EFC 360°

10.5 = 150°

C 360°

10.5 = 5C 12

25.2 = C

4. Radius of G

C = 2 r

r4.01

= 2 3.12 r25.2

= r2 3.14 r25.2

The radius of G is about 4.01.

Guided Practice



EXAMPLE 5

TRACK

The curves at the ends of the track shown are 180° arcs of circles. The radius of the arc for a runner on the red path shown is 36.8 meters. About how far does this runner travel to go once around the track? Round to the nearest tenth of a meter.

SOLUTION

The path of a runner is made of two straight sections and two semicircles. To find the total distance, find the sum of the lengths of each part.

EXAMPLE



EXAMPLE 5Find arc lengths to find distance

= 2(84.39) + 2 2π 36.812

≈ 400.0 meters

The runner on the red path travels about 400 meters.

ANSWER



GUIDED PRACTICE

5. The radius of the arc for a runner on the blue path is 44.02 meters, as shown in the diagram. About how far does this runner travel to go once around the track? Round to the nearest tenth of a meter.

= 2(84.39) + 2 2π 44.0212

≈ 445.4 meters

The runner on the blue path travels about 445.4 meters.

ANSWER

Guided Practice



Find each arc length. Give answers in terms of and rounded to the nearest hundredth.

6. FG

Use formula for area of sector.

5.96 cm 18.71 cm

Substitute 8 for r and 134 for m.

Simplify.

Guided Practice



Find each arc length. Give answers in terms of and rounded to the nearest hundredth.7. An arc with measure 62 in a circle with radius 2 m


0.69 m 2.16 m


Simplify.

Guided Practice



Find each arc length. Give your answer in terms of and rounded to the nearest hundredth.

8. GH



Simplify.= m 4.19 m

Guided Practice



Find each arc length. Give your answer in terms of and rounded to the nearest hundredth.

9. An arc with measure 135° in a circle with radius 4 cm


= 3 cm 9.42 cm


Simplify.

Guided Practice



Homework Page 226-227 (#1 – 10, 17-21 all)

circumference and arc length

Documents