Circular Measures

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Circular Measures

CHAPTER 8 : CIRCULAR MEASURES 8.1

Radian

(a)

Unit for angle (i) Degree and minute [ 1o = 60’] (ii) Radian

(b)

Radian - 1 radian ( 1 Rad.) is angle that subtended at the center of a circle by an equal in length to the radius of the circle. A r

r

O Angle AOB = 1 rad

r B

Angle of a whole turn

(c)

=

Circumfere nce of a circle Radius

=

2 r r

The relation between degree and radian.

 rad = 180o

8.1.1

2 rad

=

1 rad =

180 o





1 o=

180

rad

Converting measurements in radians to degrees and vice versa.



180 o



Radian

Degrees



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1 rad =

1 o=

 180 o

2

180 o

 180



= __________

rad = _________

Circular Measures

Exercise 1: 1. Convert each of the following angles in degrees to radians (in terms of  ) (a)

Example: 30o =

(b)

(c)

30    180 6

75o =

(d)

45o =

60o =

   4 

(e)

90o =

   3 

(f)

 5   12 

270o =  3   2 

   2 

2. Convert each of the following angles in degrees to radians Example:

(a)

15o = (d)

(b)

25o =

(c)

15  3.142  0.2618 180 65o15’=

[0.4364] (e)

150o =

26o =

(h)

95o 20’ =

115o =

[2.6183]

8.5o =

[2.0074] (i)

[0.4538] (j)

[1.3123] (f)

[1.1390] (g)

75o12’ =

325o =

[0.1484] (k)

65o 32’=

[1.6641]

[5.6731] (l)

315o =

[1.1440]

[0.5570]

3. Convert each of the following angles to degrees. (a)

 6 (d)

Example:

rad =

 6

(b)

180  30

7  rad = 4

2  rad = 3

(e)

0.6 rad =

7  rad = 6

[135°]

(f)

2  rad = 4

[210°]

(h)

[247.59°]

3

[90°]

(i)

4.32 rad =

[34.37°]

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3  rad = 4

[120°]

[315°]

(g)

(c)

5.12 rad =

[293.32°]

Circular Measures

8.2

Arc Length of a Circle A r

arc length, s  r ,

O 

 = radian

B

Exercise 1: 1. In the following circle, calculate length of arc, s.

A

s  r  5(0.9)  4.5cm

5cm O

0.9 rad

B 2. Calculate the length of arc S which subtends an angle of 20o at the center of a circle of radius 5 cm. (Give your answer correct to 2 decimal places.)

5 cm O

20O

S

[1.75cm]

3. Calculate the length of the major arc PQ of the circle below. Give the answer correct to 3 decimal places.

O 8 cm 60 Q

o

P 8 cm [41.894cm]

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Circular Measures

4. Find the length of the minor arc, S which subtends an angle of 65o 36’ at the center of a circle of radius 6 cm.

6 cm O

65o36’ S

[6.871cm]

5. The radius of its circle is 4.2 cm and the angle that subtends of the minor arc is 1.4rad. Find the length of the major arc AB. A

O

1.4 rad

B [20.513cm]

6. The length of the minor arc of the circle which radius 12 cm is 20.4 cm. Calculate the angle that subtends at the center.(Give the answer in degrees and minutes)

12 cm O



20.4 cm

[97°23’]

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Circular Measures

Exercise 2: 1. Given that the length of arc, S of the circle is 6 cm. Find its radius.

r 30o

O

S

r [11.458cm]

2. Calculate the radius of the circle below if the length of arc AB is 14 cm. A

O

2.8 rad

B [5cm]

3. Given that the length of major arc PQ is 31.5 cm. Find the angle,  below. P

31.5 cm 7 cm

 Q [102.2°]

4. Given that the length of the minor arc of the circle which subtends an angle 45o at the center is 15 cm. Find its radius. Give the answer correct to 1 decimal place.

O

45o

15 cm

[19.1cm]

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Circular Measures

8.2.1 1

Finding the perimeter of segment of a circle. Find the perimeter of the shaded segment in each of the following diagram.

Example

(a)

A

M

A

2.1rad

B

O

7cm

5cm

O

B

x

51.56

7cm

O

B

1.8rad O

B

A

A

s  r  5(2.1)  10.5cm x sin 60.15  5 x  5sin 60.15  4.3367cm AB  2 x  2(4.3367)  8.6734cm Perimeter  8.6734  10.5  23.566cm [23.566cm]

(b)

(c) B

O B

2.2rad

O O

O

115 

10cm A

A B

A

A

B

[18.434cm]

[39.82cm]

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5cm

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Circular Measures

8.3 8.3.1

Area of Sector of A Circle Determining area of a sector, radius and angle subtended at the center of a circle. P r

Q



O

A

A = Area of a sector r = radius  = angle in radian

1 2 r 2

r R

Note : 1 2

PQR is a minor arc and PSR is a major arc. The shaded part is a sector minor and the left part is a sector major.

Exercise 1 : 1

Calculate the area of the shaded sector of each of the following diagram.

Example:

(a) 4cm 1.2 rad

O

1 2 r 2 1  (4) 2 (1.2) 2  9.6cm 2

A

O

6cm

B

2.2 rad

A

[39.6cm2]

(b)

(c) 10cm

1.5 rad O

4.8rad

9cm

O

[240cm2]

[190.88cm2]

(d)

(e) B O

6cm

135 

200 

10cm

O A

\ [174.55cm2]

[42.426cm2]

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Circular Measures

Exercise 2 : 1. Solve each of the following problems: Example:

r

(a)

Given the area and the angle of the shaded region OAB is 100cm2 and B _ 3.5 rad respectively. Find r.

O

_A

A O

Given the area and the angle of the shaded region OAB is 25cm2 and 2 rad respectively. Find OA.

B

1 2 r (3.5)  100 2 100(2) r2   57.1429 3.5 r  57.1429  7.56cm

[5cm]

(b)

(c)

Given the area and the angle of the shaded region OAB is 160cm2 and 2.5 rad respectively. Find OB.

A

O B

_B



O

_A

[11.31cm]

(d)

A

[2.967rad]

(e)

Given the area of the shaded region is 140cm2 and A OB = 8 cm. Find the angle  .

 O

Given the area of the shaded region OAB is 72.7cm2 and OA=7cm. Find the angle  .

Given the area of the shaded region is 31.5 cm2 and OA = 6cm . Find the angle  .

O

B B

[4.375rad]

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[1.75rad]

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Circular Measures

8.3.2

Area of a segment

A

Area of segment

r =

=Area of AOB – Area of triangle AOB

1 2 r 2



=

1 2 r sin  2

1 2

Area of segment = r 2 (  sin  )

Exercise 1: 1 Calculate the area of the segment of each of the following. Example: (a)

45 45     0.7855 180 Area of segment 1  (6) 2 (0.7855  sin 45) 2  1.411cm 2

A 6cm 45 

O

B

A 4cm

 O

rad

3

B

[1.448cm2]

(b)

(c)

A

2

10cm

8cm



rad

O

A

O 2.4rad

B

B

[18.272cm2]

(d)

[86.21cm2]

(e)

B

A 80

8cm

A

9cm O 130 

O

B

[16.65cm2]

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[48.09cm2]

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Circular Measures

Past Year Questions:

R 1

SPM 2003(P1) Diagram shows a sector ROS with center O.



O R

  0.8153rad S S is 25 cm. Find the The length of the arc RS is 7.24 cm and the perimeter of the sector ROS

value of  , in rad. 2

[ 3 marks ]

SPM 2004 (P1) Diagram shows a circle with center O.

A O

0.354 rad

B

Given that the length of the major arc AB is 45.51 cm, find the length, in cm, of the radius.(Use  = 3.142 ) [ 3 marks ] r= 7.675 cm

3

SPM 2005 (P1) Diagram shows a circle with center O.

A

O

(a) 1.222 rad (b) r = 13.09 cm

 B

The length of the minor arc AB is 16 cm and the angle of the major sector AOB is 290o. Using  = 3.142 , find

(a)

the value of  , in radians

(b)

(Give your answer correct to four significant figures) the length, in cm, of the radius of the circle. [ 3 marks ]

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