Develop and use the formulas for the circumference, area, arc length, and sector area of circles

1. Circumference and Area of Circles
The student is able to (I can):
• Develop and use formulas to find the circumference and
area of circles
• Develop and use formulas to find arc length and sector
area of circles

2. ππππ ((((pi)pi)pi)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
2C r= π
which becomes

3. If you cut a circle into wedges, and arrange the wedges
into a parallelogram-shaped figure:
radius
1
2
circumference
A = bh
1
circumference radius
2
A = i
( )
1
2 radius radius
2
A = π i
2
A r= π
The more wedges, the closer the figure is to a parallelogram.

5. Examples
1. Find the exact circumference and area of a circle whose
diameter is 18 m.
2. Find the diameter and area of a circle whose
circumference is 22π ft.
3. Find the radius of a circle whose area is 81π cm2.

6. Examples
1. Find the exact circumference and area of a circle whose
diameter is 18 m.
C = πd = π(18) = 18π m
A = πr2 = π(92) = 81π m2
2. Find the diameter and area of a circle whose
circumference is 22π ft.
22π = πd
d = 22 ft
A = π(112) = 121π ft2
3. Find the radius of a circle whose area is 81π cm2.
81π = πr2
81 = r2
81 9 cmr = =

7. arcarcarcarc lengthlengthlengthlength – the distance along an arc. It is proportional to
the circumference of the circle.
or
360
m
L C
° 
=  
° 
where C is the circumference
(either C=πd or C=2πr).
•
mmmm°°°°
m° arc length central angle
circumference 360
360
L m
C
=
°
°
=
°

8. Example
Find each exact arc length.
1.
2.
120°
• 3333′′′′
72727272°°°°
8 m8 m8 m8 m
•

9. Example
Find each exact arc length.
1.
2.
120°
• 3333′′′′
( )
120
2 3
360
2 ft.
L
 
= π  
 
= π
( )
72
2 8
360
16
or 3.2 m
5
L
 
= π  
 
= π π
72727272°°°°
8 m8 m8 m8 m
•

10. 3. Find the radius of the circle.
•
130130130130°°°°
rrrr
39π m

11. 3. Find the radius of the circle.
•
130130130130°°°°
rrrr
39π m
39 130
2 360
260 14040
14040
54 m
260
r
r
r
π °
=
π °
π = π
π
= =
π

12. sector of asector of asector of asector of a circlecirclecirclecircle – a region bounded by a central angle. It is
proportional to the area of the circle containing the
sector.
•
R
AAAA
G
∠RAG is a central angle
RAG is a sector
Area of sector central angle
Area of circle 360
=
°
2
S
360
m
r
°
=
π °
2
360
m
S r
° 
= π  
° 
Formula:

13. Examples
Find the area of each sector. Leave answers in terms of π.
1.
2.
•
120° 2"2"2"2"
• 72°
10m10m10m10m

14. Examples
Find the area of each sector. Leave answers in terms of π.
1.
2.
•
120° 2"2"2"2"
• 72°
10m10m10m10m
( )2
2
120
2
360
4 120 480
360 360
4
in
3
S
°
= π
°
π⋅ π
= =
= π
( )2
2
72
10
360
7200
20 m
360
S
° 
= π  
° 
π
= = π