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Radians, Arc Length and Sector Area
40: Radians, Arc Length 40: Radians, Arc Length and Sector Areaand Sector Area
Radians, Arc Length and Sector Area
Radians
Radians are units for measuring angles.They can be used instead of degrees.
r
O
1 radian is the size of the angle formed at the centre of a circle by 2 radii which join the ends of an arc equal in length to the radius.
r
r
x = 1 radian
x
= 1 rad. or 1c
Radians, Arc Length and Sector Area
r
O
2r
r
2c
If the arc is 2r, the angle is 2 radians.
Radians
Radians, Arc Length and Sector Area
O
If the arc is 3r, the angle is 3 radians.
r3r
r
3c
If the arc is 2r, the angle is 2 radians.
Radians
Radians, Arc Length and Sector Area
O
If the arc is 3r, the angle is 3 radians.
c143
If the arc is 2r, the angle is 2 radians.
r
r
If the arc is r, the angle is radians.
143 143
r143
Radians
Radians, Arc Length and Sector Area
O
If the arc is 3r, the angle is 3 radians.
r
r
If the arc is 2r, the angle is 2 radians.
If the arc is r, the angle is radians.
143 143
If the arc is r, the angle is radians.
rc
Radians
Radians, Arc Length and Sector Area
If the arc is r, the angle is radians.
O
r
r
rc
But, r is half the circumference of the circle so the angle is
180
180 radians Hence,
Radians
Radians, Arc Length and Sector Area
We sometimes say the angle at the centre is subtended by the arc.
180 radians
Hence,
180
357
radian 1
r
O
r
rx
x = 1 radian357
Radians
Radians, Arc Length and Sector Area
Radians
SUMMARY
• One radian is the size of the angle subtended by the arc of a circle equal to the radius
180 radians • 1 radian = 180
degrees
1 degree = 180
radians
180 radians •
Radians, Arc Length and Sector Area
SUMMARY
1 radian = 180
degrees
1 degree = 180
radians
degrees to radians Multiply by 180
Memory Aid Dr by 180
Degrees to radians X by
180
radians to degrees Multiply by
180
Radians, Arc Length and Sector Area
Exercises
1. Write down the equivalent number of degrees for the following number of radians:
Ans:
(a) (b) (c) (d)2
3
26
(a) (b) (c) (d)60 45 120 30
2. Write down, as a fraction of , the number of radians equal to the following:
(a) (b) (c) (d)6090 360 30
(a) (b) (c) (d)3
6
32
4Ans:
It is very useful to memorize these conversions
Radians, Arc Length and Sector Area
Arc Length and Sector Area
Let the arc length be l .
O
r
r
l
rl 22
Consider a sector of a circle with angle .
θ
Then, whatever fraction is of the total angle at O, . . .
θ
θrl
2
θ. . . l is the same fraction of the circumference. So,
( In the diagram this is about one-third.)
2
l circumference
2
lcircumference
Radians, Arc Length and Sector Area
O
r
r
θ
θrA 2
2
1
Also, the sector area A is the same fraction of the area of the circle.
A
2
Acircle area
2
2rθA
Arc Length and Sector Area
Radians, Arc Length and Sector Area
Examples
1. Find the arc length, l, and area, A, of the sector of a circle of radius 7 cm. and sector angle 2 radians.
Solution: where is in radians
θrl θ
cm.14)2)(7( ll
θrA 221 .cm249)2()7(
2
1 2 AA
Radians, Arc Length and Sector Area
2. Find the arc length, l, and area, A, of the sector of a circle of radius 5 cm. and sector angle . Give exact answers in terms of .
150
Solution: where is in radians
θrl θ
rads5
150 150180 6
So, cm.6
25
6
55
llrθl
θrA 221 .cm2
12
125
6
5)5(
2
1 2
AA
Examples
Memory Aid Dr by 180
Degrees to radians X by
180
Radians, Arc Length and Sector Area
Radians• An arc of a circle equal in length to the
radius subtends an angle equal to 1 radian. 180 radians •
• 1 radian 357
θrl
θrA 221
For a sector of angle radians of a circle of radius r,
θ
• the arc length, l, is given by
• the sector area, A, is given by
SUMMARY
Radians, Arc Length and Sector Area
1. Find the arc length, l, and area, A, of the sector shown. O
4 cm
A
c2
l
2. Find the arc length, l, and area, A, of the
sector of a circle of radius 8 cm. and sector
angle . Give exact answers in terms of
.
120
Exercises
Radians, Arc Length and Sector Area
1. Solution:
θrl cm.8)2)(4( l
θrA 221 .cm216)2()4( 2
21 A
O4 cm
A
c2
l
Exercises
Radians, Arc Length and Sector Area
2. Solution:
rads2
120 120180 3
So, cm.3
16
3
28
llrθl
θrA 221 .cm2
3
64
3
2)8(
2
1 2
AA
O8 cm
A
120
l
where is in radians
θrl
Exercises
Memory Aid Dr by 180
Degrees to radians X by
180
Radians, Arc Length and Sector Area
Radians, Arc Length and Sector Area
The following slides contain repeats of information on earlier slides, shown without colour, so that they can be printed and photocopied.For most purposes the slides can be printed as “Handouts” with up to 6 slides per sheet.
Radians, Arc Length and Sector Area
Radians
SUMMARY
• One radian is the size of the angle subtended by the arc of a circle equal to the radius
180 radians •
• 1 radian 357
r
O
r
rx
Radians, Arc Length and Sector AreaArc Length and Sector
Area
Let the arc length be l .
O
r
r
l
rl 22
Consider a sector of a circle with angle .
θ
Then, whatever fraction is of the total angle at O, . . .
θ
θrl
2
θ. . . l is the same fraction of the circumference. So,
( In the diagram this is about one-third.)
2
l circumference
2
lcircumference
Radians, Arc Length and Sector Area
2
2rθA
Arc Length and Sector Area
O
r
r
θ
θrA 2
2
1
Also, the sector area A is the same fraction of the area of the circle.
A
2
Acircle area
Radians, Arc Length and Sector Area
SUMMARY
θrl
θrA 221
For a sector of angle radians of a circle of radius r,
θ
• the arc length, l, is given by
• the sector area, A, is given by
![1 CIRCULAR MOTION 2 r s IN RADIANS length of the arc [ s ] divided by the radius [ r ] subtending the arc](https://img.pdfslide.net/doc/110x75/56649e2b5503460f94b196fc/1-circular-motion-2-r-s-in-radians-length-of-the-arc-s-divided-by-the.jpg)
