Lesson 6.7
Arc Length
You have learned that the measure of a minor arc is equal to the measure of its
central angle.
On a clock, the measure of the arc from 12:00 to 4:00 is equal to the measure of the angle
formed by the hour and minute hands.
A circular clock is divided into 12 equal arcs, so the measure of each hour is
The measure of the arc from 12:00 to 4:00 is four times 30°, or 120°.
𝟑𝟔𝟎°
𝟏𝟐
, or 30°.
Notice that because the minute hand is longer, the tip of the minute hand must travel
farther than the tip of the hour hand even though they both move 120° from 12:00 to
4:00.
So the arc length is different even though the arc measure is the same!
JRLeon
Geometry Chapter 6.7
HGSH
Arc Length
Lesson 6.7
Let’s take another look at the arc measure.
EXAMPLE 1
Solution
What fraction of its circle is each arc?
𝟗𝟎°
𝟏
=
𝟑𝟔𝟎° 𝟒
𝟏𝟖𝟎° 𝟏
=
𝟑𝟔𝟎° 𝟐
𝟏𝟐𝟎° 𝟏
=
𝟑𝟔𝟎° 𝟑
What do these fractions have to do with arc length? If you traveled halfway around a
1
circle, you’d cover 2 of its perimeter, or circumference. If you went a quarter of the way
1
around, you’d travel ed 4 of its circumference. The arc length is some fraction of the
circumference of its circle. The measure of an arc is calculated in units of degrees, but arc
length is calculated in units of distance.
JRLeon
Geometry Chapter 6.7
HGSH
Arc Length
Lesson 6.7
Arc Length Conjecture
The length of an arc equals the
(
𝒎𝒆𝒂𝒔𝒖𝒓𝒆 𝒐𝒇 𝒕𝒉𝒆 𝒂𝒓𝒄
)𝑪
𝟑𝟔𝟎
C-66
Where C = d or C = 2r
EXAMPLE 2
Solution
JRLeon
Geometry Chapter 6.7
HGSH
Lesson 6.7
Arc Length
C = d or C = 2r
EXAMPLE 3
Solution
JRLeon
Geometry Chapter 6.7
HGSH
Lesson 6.7
Arc Length
C = d or C = 2r
Intersecting Secants Conjecture
The measure of an angle formed by two secants that intersect outside a
circle one-half the difference of the larger intercepted arc
measure and the smaller intercepted arc measure.
JRLeon
Geometry Chapter 6.7
HGSH
Lesson 6.7
Arc Length
LESSON 6.7 : Pages 351-352, problems 1 through 10, 15.
JRLeon
Geometry Chapter 6.5
HGSH