![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
r Angle AOB = 1 rad
B
Circumference of a circle
Angle of a whole turn =
Radius
2π r
= = 2π rad
r
(c) The relation between degree and radian.
π rad = 180o 180 o π
1 rad = 1 o= rad
π 180
8.1.1 Converting measurements in radians to degrees and vice versa.
180 o
×
π
1 rad = = __________
Radian Degrees
1 o = = _________
π
×
180 o
PPMP Matematik Tambahan 1
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![Circular Measures
Exercise 1:
1. Convert each of the following angles in degrees to radians (in terms of π)
(a) Example: (b) 45o = (c) 60o =
30° π π π
30o = ×π = 4 3
180° 6
(d) 75o = (e) 90o = (f) 270o =
5π π 3π
12
2
2
2. Convert each of the following angles in degrees to radians
(a) Example: (b) 25o = (c) 75o12’ =
15°
15o = × 3.142 = 0.2618
180° [0.4364] [1.3123]
(d) 65o15’= (e) 150o = (f) 115o =
[1.1390] [2.6183] [2.0074]
(g) 26o = (h) 8.5o = (i) 325o =
[0.4538] [0.1484] [5.6731]
(j) 95o 20’ = (k) 65o 32’= (l) 315o =
[1.6641] [1.1440] [0.5570]
3. Convert each of the following angles to degrees.
(a) Example: 2 3
(b) π rad = (c) π rad =
π π 3 4
rad = ×180° = 30°
6 6 [120°] [135°]
7 7 2
(d) π rad = (e) π rad = (f) π rad =
4 6 4
[315°] [210°] [90°]
(g) 0.6 rad = (h) 4.32 rad = (i) 5.12 rad =
[34.37°] [247.59°] [293.32°]
8.2 Arc Length of a Circle
PPMP Matematik Tambahan 2
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![Circular Measures
A
r
arc length, s = rθ , θ = radian
O θ
B
Exercise 1:
1. In the following circle, calculate length of arc, s.
A s = rθ
5cm
= 5(0.9)
O 0.9 rad = 4.5cm
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
S
O 20O
[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
60o
Q P
8 cm
[41.894cm]
PPMP Matematik Tambahan 3
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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
20.4 cm
O θ
[97°23’]
PPMP Matematik Tambahan 4
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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
S
O 30o
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, θ below.
P
31.5 cm
7 cm
θ
Q
[102.2°]
4. Given that the length of the minor arc of the circle which subtends an angle 45 o at the center is
15 cm. Find its radius. Give the answer correct to 1 decimal place.
15 cm
O 45o
[19.1cm]
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![Circular Measures
8.2.1 Finding the perimeter of segment of a circle.
1 Find the perimeter of the shaded segment in each of the following diagram.
Example (a)
A A M x B O
2.1rad B
7cm
5cm
O 51.56°
7cm
O
1.8rad B
O
A A B
s = rθ = 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 5cm
10cm 115°
A A
A B A B
[39.82cm] [18.434cm]
PPMP Matematik Tambahan 6
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![Circular Measures
8.3 Area of Sector of A Circle
8.3.1 Determining area of a sector, radius and angle subtended at the center of a circle .
P
A = Area of a sector
r Q r = radius
1 2 = angle in radian
A= rθ
O θ 2
r
R
Note :
1 PQR is a minor arc and PSR is a major arc.
2 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)
1
A = r 2θ
4cm 2
O
1 2 6cm
O 1.2 rad = (4) (1.2) B
2 A 2.2 rad
= 9.6cm 2
[39.6cm2]
(b) (c)
10cm
1.5πrad
O
4.8rad
9cm O
[240cm2] [190.88cm2]
(d) (e)
B
6cm O 200°
135°
O 10cm
A
[42.426cm2] [174.55cm2]
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![Circular Measures
Exercise 2 :
1. Solve each of the following problems:
Example: Given the area and the (a) Given the area and the
angle of the shaded angle of the shaded
region OAB is 100cm2 region OAB is 25cm2
r O A
and and
B
_
A
_ 3.5 rad respectively. O 2 rad respectively.
Find r. Find OA.
1 2 B
r (3.5) = 100
2
100(2)
r2 = = 57.1429
3.5
r = 57.1429 = 7.56cm
[5cm]
(b) Given the area and the (c) Given the area of the
A angle of the shaded B
_ shaded region OAB is
region OAB is 72.7cm2 and
O
160cm2 and 2.5 rad
respectively. θO OA=7cm.
Find the angle θ .
Find OB.
A
_
B
[11.31cm] [2.967rad]
(d) (e)
Given the area of Given the area of the
the shaded region is shaded region is 31.5
140cm2 and OB = 8 A cm2 and OA = 6cm .
θ cm.
O
Find the angle θ .
A O Find the angle θ .
B
B
[4.375rad]
[1.75rad]
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![Circular Measures
8.3.2 Area of a segment
A Area of segment =Area of AOB – Area of triangle AOB
r
1 2 1 2
= r θ − r sin θ
2 2
=
1 2
Area of segment = r (θ − sin θ )
2
Exercise 1:
1 Calculate the area of the segment of each of the following.
Example: (a)
45° A
A
45° ⇒ × π = 0.7855
180° 4cm
6cm π
Area of segment 3
rad
45° O O
1
= (6) 2 (0.7855 − sin 45°) B
2
B
= 1.411cm 2
[1.448cm2]
(b) (c)
A
A
8cm 10cm
π O
rad O 2.4rad
2
B B
[18.272cm2] [86.21cm2]
(d) (e)
B A A
8cm
80°
9cm
O 130°
O
B
[16.65cm2]
[48.09cm2]
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![Circular Measures
Past Year Questions:
R
1 SPM 2003(P1)
Diagram shows a sector ROS with center O.
θ
OR SS θ = 0.8153rad
The length of the arc RS is 7.24 cm and the perimeter of the sector ROS is 25 cm. Find the
value of θ , in rad. [ 3 marks ]
2 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 π = 3.142 ) [ 3 marks ]
r= 7.675 cm
3 SPM 2005 (P1)
Diagram shows a circle with center O.
A
θ (a) 1.222 rad
O (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 π = 3.142 , find
(a) the value of θ , in radians
(Give your answer correct to four significant figures)
(b) the length, in cm, of the radius of the circle. [ 3 marks ]
PPMP Matematik Tambahan 10
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![Circular Measures
Past Year Questions:
R
1 SPM 2003(P1)
Diagram shows a sector ROS with center O.
θ
OR SS θ = 0.8153rad
The length of the arc RS is 7.24 cm and the perimeter of the sector ROS is 25 cm. Find the
value of θ , in rad. [ 3 marks ]
2 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 π = 3.142 ) [ 3 marks ]
r= 7.675 cm
3 SPM 2005 (P1)
Diagram shows a circle with center O.
A
θ (a) 1.222 rad
O (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 π = 3.142 , find
(a) the value of θ , in radians
(Give your answer correct to four significant figures)
(b) the length, in cm, of the radius of the circle. [ 3 marks ]
PPMP Matematik Tambahan 10
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![Circular Measures
Past Year Questions:
R
1 SPM 2003(P1)
Diagram shows a sector ROS with center O.
θ
OR SS θ = 0.8153rad
The length of the arc RS is 7.24 cm and the perimeter of the sector ROS is 25 cm. Find the
value of θ , in rad. [ 3 marks ]
2 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 π = 3.142 ) [ 3 marks ]
r= 7.675 cm
3 SPM 2005 (P1)
Diagram shows a circle with center O.
A
θ (a) 1.222 rad
O (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 π = 3.142 , find
(a) the value of θ , in radians
(Give your answer correct to four significant figures)
(b) the length, in cm, of the radius of the circle. [ 3 marks ]
PPMP Matematik Tambahan 10
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![Circular Measures
Past Year Questions:
R
1 SPM 2003(P1)
Diagram shows a sector ROS with center O.
θ
OR SS θ = 0.8153rad
The length of the arc RS is 7.24 cm and the perimeter of the sector ROS is 25 cm. Find the
value of θ , in rad. [ 3 marks ]
2 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 π = 3.142 ) [ 3 marks ]
r= 7.675 cm
3 SPM 2005 (P1)
Diagram shows a circle with center O.
A
θ (a) 1.222 rad
O (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 π = 3.142 , find
(a) the value of θ , in radians
(Give your answer correct to four significant figures)
(b) the length, in cm, of the radius of the circle. [ 3 marks ]
PPMP Matematik Tambahan 10
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add mad
- 1. 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 r Angle AOB = 1 rad B Circumference of a circle Angle of a whole turn = Radius 2π r = = 2π rad r (c) The relation between degree and radian. π rad = 180o 180 o π 1 rad = 1 o= rad π 180 8.1.1 Converting measurements in radians to degrees and vice versa. 180 o × π 1 rad = = __________ Radian Degrees 1 o = = _________ π × 180 o PPMP Matematik Tambahan 1 Negeri Johor
- 2. Circular Measures Exercise 1: 1. Convert each of the following angles in degrees to radians (in terms of π) (a) Example: (b) 45o = (c) 60o = 30° π π π 30o = ×π = 4 3 180° 6 (d) 75o = (e) 90o = (f) 270o = 5π π 3π 12 2 2 2. Convert each of the following angles in degrees to radians (a) Example: (b) 25o = (c) 75o12’ = 15° 15o = × 3.142 = 0.2618 180° [0.4364] [1.3123] (d) 65o15’= (e) 150o = (f) 115o = [1.1390] [2.6183] [2.0074] (g) 26o = (h) 8.5o = (i) 325o = [0.4538] [0.1484] [5.6731] (j) 95o 20’ = (k) 65o 32’= (l) 315o = [1.6641] [1.1440] [0.5570] 3. Convert each of the following angles to degrees. (a) Example: 2 3 (b) π rad = (c) π rad = π π 3 4 rad = ×180° = 30° 6 6 [120°] [135°] 7 7 2 (d) π rad = (e) π rad = (f) π rad = 4 6 4 [315°] [210°] [90°] (g) 0.6 rad = (h) 4.32 rad = (i) 5.12 rad = [34.37°] [247.59°] [293.32°] 8.2 Arc Length of a Circle PPMP Matematik Tambahan 2 Negeri Johor
- 3. Circular Measures A r arc length, s = rθ , θ = radian O θ B Exercise 1: 1. In the following circle, calculate length of arc, s. A s = rθ 5cm = 5(0.9) O 0.9 rad = 4.5cm 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 S O 20O [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 60o Q P 8 cm [41.894cm] PPMP Matematik Tambahan 3 Negeri Johor
- 4. 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 20.4 cm O θ [97°23’] PPMP Matematik Tambahan 4 Negeri Johor
- 5. Circular Measures Exercise 2: 1. Given that the length of arc, S of the circle is 6 cm. Find its radius. r S O 30o 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, θ below. P 31.5 cm 7 cm θ Q [102.2°] 4. Given that the length of the minor arc of the circle which subtends an angle 45 o at the center is 15 cm. Find its radius. Give the answer correct to 1 decimal place. 15 cm O 45o [19.1cm] PPMP Matematik Tambahan 5 Negeri Johor
- 6. Circular Measures 8.2.1 Finding the perimeter of segment of a circle. 1 Find the perimeter of the shaded segment in each of the following diagram. Example (a) A A M x B O 2.1rad B 7cm 5cm O 51.56° 7cm O 1.8rad B O A A B s = rθ = 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 5cm 10cm 115° A A A B A B [39.82cm] [18.434cm] PPMP Matematik Tambahan 6 Negeri Johor
- 7. Circular Measures 8.3 Area of Sector of A Circle 8.3.1 Determining area of a sector, radius and angle subtended at the center of a circle . P A = Area of a sector r Q r = radius 1 2 = angle in radian A= rθ O θ 2 r R Note : 1 PQR is a minor arc and PSR is a major arc. 2 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) 1 A = r 2θ 4cm 2 O 1 2 6cm O 1.2 rad = (4) (1.2) B 2 A 2.2 rad = 9.6cm 2 [39.6cm2] (b) (c) 10cm 1.5πrad O 4.8rad 9cm O [240cm2] [190.88cm2] (d) (e) B 6cm O 200° 135° O 10cm A [42.426cm2] [174.55cm2] PPMP Matematik Tambahan 7 Negeri Johor
- 8. Circular Measures Exercise 2 : 1. Solve each of the following problems: Example: Given the area and the (a) Given the area and the angle of the shaded angle of the shaded region OAB is 100cm2 region OAB is 25cm2 r O A and and B _ A _ 3.5 rad respectively. O 2 rad respectively. Find r. Find OA. 1 2 B r (3.5) = 100 2 100(2) r2 = = 57.1429 3.5 r = 57.1429 = 7.56cm [5cm] (b) Given the area and the (c) Given the area of the A angle of the shaded B _ shaded region OAB is region OAB is 72.7cm2 and O 160cm2 and 2.5 rad respectively. θO OA=7cm. Find the angle θ . Find OB. A _ B [11.31cm] [2.967rad] (d) (e) Given the area of Given the area of the the shaded region is shaded region is 31.5 140cm2 and OB = 8 A cm2 and OA = 6cm . θ cm. O Find the angle θ . A O Find the angle θ . B B [4.375rad] [1.75rad] PPMP Matematik Tambahan 8 Negeri Johor
- 9. Circular Measures 8.3.2 Area of a segment A Area of segment =Area of AOB – Area of triangle AOB r 1 2 1 2 = r θ − r sin θ 2 2 = 1 2 Area of segment = r (θ − sin θ ) 2 Exercise 1: 1 Calculate the area of the segment of each of the following. Example: (a) 45° A A 45° ⇒ × π = 0.7855 180° 4cm 6cm π Area of segment 3 rad 45° O O 1 = (6) 2 (0.7855 − sin 45°) B 2 B = 1.411cm 2 [1.448cm2] (b) (c) A A 8cm 10cm π O rad O 2.4rad 2 B B [18.272cm2] [86.21cm2] (d) (e) B A A 8cm 80° 9cm O 130° O B [16.65cm2] [48.09cm2] PPMP Matematik Tambahan 9 Negeri Johor
- 10. Circular Measures Past Year Questions: R 1 SPM 2003(P1) Diagram shows a sector ROS with center O. θ OR SS θ = 0.8153rad The length of the arc RS is 7.24 cm and the perimeter of the sector ROS is 25 cm. Find the value of θ , in rad. [ 3 marks ] 2 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 π = 3.142 ) [ 3 marks ] r= 7.675 cm 3 SPM 2005 (P1) Diagram shows a circle with center O. A θ (a) 1.222 rad O (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 π = 3.142 , find (a) the value of θ , in radians (Give your answer correct to four significant figures) (b) the length, in cm, of the radius of the circle. [ 3 marks ] PPMP Matematik Tambahan 10 Negeri Johor
- 11. Circular Measures Past Year Questions: R 1 SPM 2003(P1) Diagram shows a sector ROS with center O. θ OR SS θ = 0.8153rad The length of the arc RS is 7.24 cm and the perimeter of the sector ROS is 25 cm. Find the value of θ , in rad. [ 3 marks ] 2 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 π = 3.142 ) [ 3 marks ] r= 7.675 cm 3 SPM 2005 (P1) Diagram shows a circle with center O. A θ (a) 1.222 rad O (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 π = 3.142 , find (a) the value of θ , in radians (Give your answer correct to four significant figures) (b) the length, in cm, of the radius of the circle. [ 3 marks ] PPMP Matematik Tambahan 10 Negeri Johor
- 12. Circular Measures Past Year Questions: R 1 SPM 2003(P1) Diagram shows a sector ROS with center O. θ OR SS θ = 0.8153rad The length of the arc RS is 7.24 cm and the perimeter of the sector ROS is 25 cm. Find the value of θ , in rad. [ 3 marks ] 2 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 π = 3.142 ) [ 3 marks ] r= 7.675 cm 3 SPM 2005 (P1) Diagram shows a circle with center O. A θ (a) 1.222 rad O (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 π = 3.142 , find (a) the value of θ , in radians (Give your answer correct to four significant figures) (b) the length, in cm, of the radius of the circle. [ 3 marks ] PPMP Matematik Tambahan 10 Negeri Johor
- 13. Circular Measures Past Year Questions: R 1 SPM 2003(P1) Diagram shows a sector ROS with center O. θ OR SS θ = 0.8153rad The length of the arc RS is 7.24 cm and the perimeter of the sector ROS is 25 cm. Find the value of θ , in rad. [ 3 marks ] 2 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 π = 3.142 ) [ 3 marks ] r= 7.675 cm 3 SPM 2005 (P1) Diagram shows a circle with center O. A θ (a) 1.222 rad O (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 π = 3.142 , find (a) the value of θ , in radians (Give your answer correct to four significant figures) (b) the length, in cm, of the radius of the circle. [ 3 marks ] PPMP Matematik Tambahan 10 Negeri Johor