Relation Between Central Angle
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RELATION BETWEEN THE CENTRAL ANGLE,
SECTOR AREA, AND THE LENGHT OF ARCSBETWEEN TWO SECTORS
Look at the
picture
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RELATION BETWEEN THE CENTRAL ANGLE,THE LENGHT OF ARCS, and SECTOR AREA
central AOB=
The lenght of
arc AB=
The area of
sector OAB
central COD
The lenght of
arc CD
The area of
sector OCD
Look at the
picture
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If central angle is compared to a full angle of circle
that is ( 3600), then :
O
central AOB=
The lenght of
arc AB=
The area of
sector OAB
3600
Circumference
of circle Area of circle
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If the central angle that be compared to the full
angle of circle ( 3600), then :
O
central AOB =
The lenght of
arc AB =
the area of
sector OAB
3600 2r r2
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Example 1 :
O
D
C
A
B
50
0
800
Look at the picture beside,
the lenght of arc AB = 40
cm, AOB = 500, and
AOB=800.
Find the lenght of arc CD.
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Solution :Given :
AB = 40 cm, AOB = 500, AOB=800
central AOB=
The lenght of arc AB
central COD The lenght of arc CD
500=
40 cm
800 X cm
X = ( 40 x 80 ) : 50 = 64.
O
D
C
A
B
500
800
Asking : Find the lenght of arc CD
Answer:
So, we get the lenght of arc CD = 64 cm
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Given :
AOB = 540, and radii = 20 cm
central AOB=
The area of sector OAB
3600
The area of circle540
=x
3600 r2
X = ( 3 x 1256 ) : 20 = 188,4 cm2.
So, we get the area of sector OAB = 188,4 cm2.
3=
x
20 3,14 x 202
Solution :
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Question - 1
Look at the picture, the
lenght of arc PQ = 50 cm,
the lenght of arc QR = 75
cm and POQ = 450.
Determine QOR.
O
P
R
450
Q
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Given :
The lenght of arc PQ = 50 cmThe lenght of arc QR = 75 cm
POQ = 450
central POQ=
The lenght of arc PQ
central QOR The lenght of arc QR
45=
50
x 75
X = ( 3 x 45) : 2 = 135 : 2 = 67,50
So, we get QOR = 67,50.
45=
2
x 3
Solution :
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Question - 2
Look at the picture,
COD = 600, the lenght of
OA = 12 cm and AC = 12 cm.
Calculate the area of the
shaded region.
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Given :
Radii (1) = 12 cm
Radii (2) = 24 cm.
AOB = 600
Solution :
Answer:Area of sector OAB = 60/360 x area of circle
= 1/6 x 3,14 x 12 x 12
= 3,14 x 24= 75,36 cm2
Asking :the area of the shadedregion.
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Area of sector OCD = 60/360 x area of circle
= 1/6
x 3,14 x 24 x 24
= 3,14 x 96
= 301,44 cm2
The area of the shaded region:
= area of sector OCD - area of sector OAB
= 301,44 cm
2
- 75,36 cm
2
= 226,08 cm2.
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Question - 3
42 cm
Calculate the area of the shaded
region as shown in the figure above !
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Solution:
The area of circle that
is shaded :
L = r2
= x 22/7 x 21 x 21
= x 22 x 63
= 11 x 63
= 693 cm2
42 cm
The area of small circle that is shaded =
The area of small circle that isnt shaded.
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Quiz - 1
Look at the picture, the area of
sector OAB =60 cm2,
AOB = 400 and BOC = 120o
Find the area of sector OBC.
O
A
C
1200
400
B
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Quiz - 2
Calculate the area of the shaded
region as shown in the figure above !
14 cm
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Given : AOB = 400 and BOC = 1200
The area of sector OAB = 60 cm2
Asking: The area of sector OBC.
Answer:
central AOB=
The area of sector OAB
central BOC The area of sector OBC
400=
60
1200
x
X = 3 x 60 = 180 cm2.
So we get the area of sector OBC = 180 cm2.
1=
60
3 x
Solution Quiz 1 :
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Solution Quiz 2:
The area of circle thatis shaded :
Lb = r2
= 22/7 x 7 x 7
= 154 cm2
Lk = r2
= 22/7 x 3,5 x 3,5
= 38,5 cm2
The area of the shaded region = 154 - 38,5
= 115,5 cm2
14 cm
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