MM2G3 Students will understand properties of circles.
Circumference and
Arc Length
Wednesday, March 23, 2016
Lesson 6.7
How do we find the
circumference of a circle and
the length of an arc of a circle?
M2 Unit 3: Day 9
MM2G3 c Use the properties of circles to solve problems involving the length of an arc and the area of a sector.
MM2G3 Students will understand properties of circles.
Warm Ups
1. Find w, y, and z. Give the answers in
simplest radical form.
MM2G3 c Use the properties of circles to solve problems involving the length of an arc and the area of a sector.
MM2G3 Students will understand properties of circles.
Warm Ups
Warm Up
Find each measure.
2. RT
54
3. UQ
MM2G3 c Use the properties of circles to solve problems involving the length of an arc and the area of a sector.
MM2G3 Students will understand properties of circles.
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
MM2G3 c Use the properties of circles to solve problems involving the length of an arc and the area of a sector.
MM2G3 Students will understand properties of circles.
Helpful Hint
The  key gives the best possible
approximation for  on your calculator.
Always wait until the last step to round.
MM2G3 c Use the properties of circles to solve problems involving the length of an arc and the area of a sector.
MM2G3 Students will understand properties of circles.
Use the formula for circumference
EXAMPLE
Find the indicated measures.
a.
Circumference of a circle with radius 9 centimeters
SOLUTION
a. C = 2 πr
=2 π 9
= 18
56.55
Write circumference formula.
Substitute 9 for r.
Simplify.
Use a calculator.
ANSWER
The circumference is about 56.55 centimeters.
MM2G3 c Use the properties of circles to solve problems involving the length of an arc and the area of a sector.
MM2G3 Students will understand properties of circles.
EXAMPLE
Use the formula for circumference
Find the indicated measures.
b.
Radius of a circle with circumference 26 meters
C = 2 πr
26 = 2 πr
26
= r
2π
4.14
r
Write circumference formula.
Substitute 26 for c.
Divide each side by 2π.
Use a calculator.
ANSWER
The radius is about 4.14 meters.
MM2G3 c Use the properties of circles to solve problems involving the length of an arc and the area of a sector.
MM2G3 Students will understand properties of circles.
GUIDED PRACTICE
Guided Practice
1. Find the circumference of a circle with
diameter 5 inches. Find the diameter of a circle
with circumference 17 feet.
SOLUTION
C=
=5
d
Write circumference formula.
Substitute 5 for d.
15.71
Use a calculator.
The circumference is about 15.71 in.
MM2G3 c Use the properties of circles to solve problems involving the length of an arc and the area of a sector.
MM2G3 Students will understand properties of circles.
GUIDED PRACTICE
1. Find the circumference of a circle with
diameter 5 inches. Find the diameter of a circle
with circumference 17 feet.
circumference = 17 feet
C =
d
d
17 =
17 = d
Write circumference formula.
Substitute 17 for c.
5.41
Use a calculator.
r
Divide each side by π.
The Diameter is about 5.41 feet.
MM2G3 c Use the properties of circles to solve problems involving the length of an arc and the area of a sector.
MM2G3 Students will understand properties of circles.
The arc length of a circle is a fraction of the
circumference of the circle.
MM2G3 c Use the properties of circles to solve problems involving the length of an arc and the area of a sector.
MM2G3 Students will understand properties of circles.
EXAMPLE 3
Find arc lengths
EXAMPLE
Find the length of each red arc.
a.
SOLUTION
°
60
a. Arc length of AB =
360°
2π(8) ≈ 8.38 centimeters
MM2G3 c Use the properties of circles to solve problems involving the length of an arc and the area of a sector.
MM2G3 Students will understand properties of circles.
EXAMPLE 3
Find arc lengths
EXAMPLE
Find the length of each red arc.
b.
SOLUTION
°
60
b. Arc length of EF =
360°
2π(11) ≈ 11.52 centimeters
MM2G3 c Use the properties of circles to solve problems involving the length of an arc and the area of a sector.
MM2G3 Students will understand properties of circles.
EXAMPLE 3
Find arc lengths
EXAMPLE
Find the length of each red arc.
c.
SOLUTION
c. Arc length of GH
120°
=
360°
2π(1) ≈ 23.04 centimeters
MM2G3 c Use the properties of circles to solve problems involving the length of an arc and the area of a sector.
MM2G3 Students will understand properties of circles.
EXAMPLE 4
EXAMPLE
Find arc lengths to find measures
Find the indicated measure.
d. Circumference C of
Z
SOLUTION
d. Arc length of XY
= m XY
360°
C
4.19 = 40°
360°
C
4.19
1
= 19
C
37.71 = C
MM2G3 c Use the properties of circles to solve problems involving the length of an arc and the area of a sector.
MM2G3 Students will understand properties of circles.
EXAMPLE 4
EXAMPLE
Find arc lengths to find measures
Find the indicated measure.
e. m RS
SOLUTION
e. Arc length of RS
2 r
360°
= m RS°
360
2
44
m RS
=
(15.28)
360°
2
44
= m RS
(15.28)
165°
MM2G3 c Use the properties of circles to solve problems involving the length of an arc and the area of a sector.
m RS
MM2G3 Students will understand properties of circles.
GUIDED PRACTICE
Guided Practice
Find the indicated measure.
2. Length of PQ
°
75
Arc length of PQ =
360°
(9)
5.89 yd
MM2G3 c Use the properties of circles to solve problems involving the length of an arc and the area of a sector.
MM2G3 Students will understand properties of circles.
GUIDED PRACTICE
Guided Practice
3. Circumference of
N
Arc length of LM = m LM
360°
C
61.26 = 270°
360°
C
61.26 = 27
36
C
81.68 m = C
MM2G3 c Use the properties of circles to solve problems involving the length of an arc and the area of a sector.
MM2G3 Students will understand properties of circles.
GUIDED PRACTICE
Guided Practice
4. Radius of G
Arc length of EF = m EF
360°
C
10.5 = 150°
360°
C
10.5 = 5
12
C
25.2 = C
C=2
r
25.2 = 2 3.12 r
2 3.14 r = r
25.2
The radius of
G is about 4.01.
4.01
r
MM2G3 c Use the properties of circles to solve problems involving the length of an arc and the area of a sector.
MM2G3 Students will understand properties of circles.
EXAMPLE 5
EXAMPLE
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.
MM2G3 c Use the properties of circles to solve problems involving the length of an arc and the area of a sector.
MM2G3 Students will understand properties of circles.
EXAMPLE 5
Find arc lengths to find distance
= 2(84.39) + 2
1
2
2π
36.8
≈ 400.0 meters
ANSWER
The runner on the red path travels about 400 meters.
MM2G3 c Use the properties of circles to solve problems involving the length of an arc and the area of a sector.
MM2G3 Students will understand properties of circles.
GUIDED PRACTICE
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
1
2
2π 44.02
≈ 445.4 meters
ANSWER
The runner on the blue path travels about 445.4 meters.
MM2G3 c Use the properties of circles to solve problems involving the length of an arc and the area of a sector.
MM2G3 Students will understand properties of circles.
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.
Substitute 8 for r and
134 for m.
 5.96 cm  18.71 cm Simplify.
MM2G3 c Use the properties of circles to solve problems involving the length of an arc and the area of a sector.
MM2G3 Students will understand properties of circles.
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
Use formula for area
of sector.
Substitute 2 for r and
62 for m.
 0.69 m  2.16 m Simplify.
MM2G3 c Use the properties of circles to solve problems involving the length of an arc and the area of a sector.
MM2G3 Students will understand properties of circles.
Guided Practice
Find each arc length. Give your answer in terms
of  and rounded to the nearest hundredth.
8. GH
Use formula for area
of sector.
Substitute 6 for r and
40 for m.
=
 m  4.19 m
Simplify.
MM2G3 c Use the properties of circles to solve problems involving the length of an arc and the area of a sector.
MM2G3 Students will understand properties of circles.
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
Use formula for area
of sector.
Substitute 4 for r and
135 for m.
= 3 cm  9.42 cm
Simplify.
MM2G3 c Use the properties of circles to solve problems involving the length of an arc and the area of a sector.
MM2G3 Students will understand properties of circles.
Homework
Page 226-227 (#1 – 10, 17-21 all)
MM2G3 c Use the properties of circles to solve problems involving the length of an arc and the area of a sector.