This document defines key concepts and formulas related to circles, including: - Circumference formulas using pi, diameter, and radius - Arc length formulas using circumference, central angle, and 360 degrees - Definition of a radian as the ratio of arc length to radius - Examples of calculating circumference, arc length, central angles, and conversions between degrees and radians

1. 6.15.1 Circumference and Arc Length
The student is able to (I can):
• Develop and use formulas for the circumference of circles
• Develop and use formulas to find arc length of circles
• Develop the definition of a radian

2. π (pi) The ratio of the circumference to the
diameter.
Since the diameter is twice the radius, this
formula can also be written as:
π is an irrational number — it never repeats
and it never ends. The symbol π is an exact
number; 3.1415926… is an approximation.
C
d
π =
C d= π
C 2 r= π
which becomes

3. Examples 1. Find the exact circumference of a circle
whose diameter is 18 in.
C = πd = π(18) = 18π in
2. Find the radius of a circle whose
circumference is 22π cm.
22π = 2πr
r = 11 cm

4. arc measure
arc length
The measure of the central angle that
intercepts the arc.
The distance along an arc. It is
proportional to the circumference of the
circle.
•
mºmºmºmº
m
L C
360
° =  
° 
where C is the
circumference (either
C=πd or C=2πr).
mº
arc length central angle
circumference 360
=
°

5. Example Find each exact arc length.
1.
2.
120º
• 3333′′′′
72º72º72º72º
8 m8 m8 m8 m
( )
120
L 2 3
360
 = π  
 
2 ft.= π
( )
72
L 2 8
360
 = π  
 
16
or 3.2 m
5
= π π
•

6. 3. Find the measure of the angle.
•
6 cm
4π cm
( )
π
=
π
π
=
π
 
= = ° 
 
4 m
2 6 360
4 m
12 360
4
m 360 120
12

7. What angle would produce an arc that
measures 10 inches on a circle with radius
of 10 inches?
Since the 10s cancel each other out, this is
true for any arc that is the same length as
its radius, and in fact, this relationship has
a special name: radianradianradianradian
( )
°
=
π
 
= = ° 
π π 
≈ °
10 m
2 10 360
360 180
m
2
57.3

8. radian The radian measure of a central angle is
the ratio of the length of the arc it
intercepts divided by the radius of the
circle.
To convert between degrees and radians:
(put the number that goes with whatever
unit you are converting totototo on toptoptoptop)

9. Examples 1. Convert 60˚ to radians
2. Convert to degrees
π
6
π π π 
° = = 
 
60
60 radians
180 180 3
π π  
= = °  
π π  
180 180
30
6 6