G.C.5 STUDENT NOTES WS #2
1
Radians and Arc Length
The arc length of an arc is the physical distance along the circumference between the two points of the arc.
We know that the full arc length of a circle is known as its circumference. So to calculate a portion of that
circumference we calculate a percentage of the circumference. If the arc is measured in degrees we get the
 θ 
following relationship, arc length = s = 
 2π r (% of the circumference). Lowercase s is often used the
 360° 
variable used to symbolize arc length.
s
60°
5 cm
 θ 
 60°
arc length = s = 
 2π r = 
 360° 
 360°
10π 5π

1
=
≈ 5.24 cm
 2π (5) =   (10π ) =
6
3

6
 θ 
 90°
arc length = s = 
 2π r = 
 360° 
 360°
32π

1
= 8π ≈ 25.13 cm
 2π (16) =   (32π ) =
4

4
s
90°
16 cm
Now if the angle is measured in terms of radians instead of degrees a nice thing happens to the formula…..
 θ 
arc length = s = 
 2π r = θ r (% of the circumference). Look how simple the formula for arc length is now
 2π 
when our angle is measured in radians, s = Ɵr.
Arc Length, s = Ɵr, when the angle is measured in radians.
π
s
 π  5π
arc length = s = θ r =   5 =
≈ 5.24 cm
3
3
3
5 cm
s
π
2
16 cm
π 
arc length = s = θ r =  16 = 8π ≈ 25.13 cm
2
Example #1
What is the length of an arc if the circle has a
radius of 4 cm and the central angle is 120°?
8
 120 
s= 
 2 π (4) = π cm ≈ 8.38 cm
3
 360 
Example #2
What the length of the an arc if the circle has a
4π
radius of 10 cm and the central angle is
?
5
 4π 
s = Ɵr = 
 10 = 8π cm ≈ 25.13 cm
 5 
G.C.5 WORKSHEET #2
Name: ________________________________ Period ______
1
1. Determine the arc length.
a) Central Angle of 30°°,
radius of 3 cm
b) Central Angle of 90°°,
radius of 8 cm
c) Central Angle of 72°°,
radius of 10 cm
s = ____________ (E)
s = ____________ (E)
s = ____________ (E)
d) Central Angle of
π
4
rad . ,
e) Central Angle of
2π
rad . ,
3
f) Central Angle of
4π
rad . ,
5
radius of 12 cm
radius of 15 cm
radius of 10 cm
s = ____________ (E)
s = ____________ (E)
s = ____________ (E)
2. After class Angela says, “I didn’t understand how he got the formula for arc length, s = Ɵr. Did you
understand it?” Explain to Angela where the formula comes from.
3. Determine the arc length of the following.
a)
b)
4π rad.
5
10 cm
s = ____________ (E)
c)
7π rad.
6
3 cm
s = ____________ (E)
d)
π rad.
6
3π rad.
2
4 cm
s = ____________ (E)
18 cm
s = ____________ (E)
4. Circle G has a radius of 7 cm. After computing an arc on circle G Nancy finds the arc length to be 14 cm.
She exclaims, “The central angle must be 2 radians.” How did she know this?
G.C.5 WORKSHEET #2
2
5. Determine the missing information.
π
a) s = 4π cm, r = 8 cm
b) Ɵ = 0.8 rad., s = 8 cm
c) r = 4.5 cm, Ɵ =
Ɵ = ____________ rad.
r = ___________ cm
s = ___________ cm
e) s = 10π cm, r = 8 cm
f) Ɵ =
Ɵ = ____________ rad.
r = ___________ cm
2π
rad., s = 5π cm
5
g) r = 8 cm, Ɵ =
π
2
3
rad.,
rad.,
s = ___________ cm
d) Ɵ =
7π
rad., s = 28π cm
4
r = ___________ cm
h) Ɵ =
5π
rad., s = 10π cm
6
r = ___________ cm
6. Find the radius of a circle in which a central angle of 5 radians intercepts an arc length of 62.5 feet?
7. Find the measure (in radians) of a central angle that intercepts an arc of length 16 cm
in a circle of radius 8 cm.
8. Find the measure (in radians) of a central angle that intercepts an arc of length 24π
π cm
in a circle of radius 10 cm.
Nome: f141y
G.C.s WORKSHEET #2
period
1. Determine the arc length.
a) Central Angle of 30o,
radius of 3 cm
y'=
b) CentralAngle of 90",
radius of 8 cm
c) Central Angle of 72",
radius of 10 cm
9r6-A-rGl)
s= ?* bq('q)
dc,^
= [*'
fuo
,= {,q
Jbo
gy
d) CentralAngle
s=
ol L rad.,
(E)
radius of 15 cm
radius of 12 cm
zd
: -7
t^*
2. After
S= tU
(E)
it?"
,o4.,
f) CentratAngle
o1?
0r)
A,
Cvlr\ lEl
s=
(E)
"l didn't understand how
Anc t"%r^ -tg a_ pzvcaa4aT.-.
o+
(r<,) =
e
S. Ll
,o4.,
radius of 10 cm
he got the formula for arc length, s =
Explain to Angela where the formula comes from.
class Angela says,
understand
(E)
$r
a-
s=
ol4
e) CentratAngle
= Vt ctrx
s
ccTcD
Or. Did you
a(-"*-<-e''
9r-
3. Determine the arc length of the following.
b)
a)
d)
c)
s=er,'+5d
ra{=
1
-_
10 cm
\
I
t..'
\ *,-/
- -/
'ry b)
s=
(E)
4. Circle G has a radius of 7
r= *rc^^-
(r)
cm. After computing
She exclaims, 'The central angle must be 2
s=
5{rc)
- .-"
(E) r= 3lf
Cn\1s1
an arc on circle G Nancy finds the arc length to be 14 cm.
radians." How did she know this?
6.C.5 WORKSHEET #2
5. Determine the missi,ng information.
als=4ncm, r=8
cm
'b)O =0.8 rad., s =8
1T
cm
c)
r=4.5cm,O=-*rad.,
9=9r-
9=*r-
tllt = eG)
"r=
4Y =0
6
Q=
v
,2r
-
f)
lr-
e=+rad.,s=5ncm
g)
gv5=
"e
=
lfi=O
?
6.
!
1f
ecr)
s= 4T
h)O=
9
'r^a.
5
a -'{
$=62,T{+
,^
r=
lengh of 62.5 feet?
=9v-
6z'<:5t
V;W
Find the measure (in radians) of a central angle
in a circle of radius 8 cm.
7.
-
that intercepts an arc of length 16 cm
g. Find the measure (in radians) of a central angle that intercepts an arc of length24n cm
in a circle of radius 10 cm.
$ = 21'1(
f=
(o
cvv..'
5- en
ct,,.':
1on cm
:47
Find the radius of a circle in which a central angle of 5 radians intercepts an arc
L-/J,
,^a.,'r=
(t)b)
.L/(
,='
r=
4
cm
r=8cm,O=-i-rad.,
3 =&r
[o7T=
7L ,^a.,s = 28n cm
k'')
5= [') l(
rad. rl
e)s=10ncm,r=8cm
(9)
d) o =
7vY
ry
-=
'saa)
e