7 Circular Measures

7/28/2019 7 Circular Measures

1/9

CIRCULAR MEASURE

1. RADIAN

1.1 Converting Measurements in Radians to Degrees and Vice Versa

Definition:r

S=rad

where Sis the length of the arcAB,ris the radius of the circle, and

is the angle subtended at the centre O by the arcAB.

For the full circle, the circumference is 2rand the angle is 360o.

rad2

2Therefore,

=

=

r

r

.

Hence, 2rad = 360o => rad = 180o

For each of the following measurement in degree, change it to radian.

Example: 25.8o

Since 180o = rad

rad4504.0

1808.258.25

=

=

(a) 30o

=

[0.5237 rad]

(b) 45o =

[0.7855 rad]

(c) 81.5o =

[1.423 rad]

(d) 122.8o =

[2.144 rad]

(e) 154o 20 =

[2.694 rad]

(f) 225o 36 =

[3.938 rad]

(g) -50o =

[-0.02773 rad]

Circular Measure 1

25.8o

O B

O B

r

r

S


2/9

For each of the following measurement in radian, change it to degree.

Example: 0.75 rad

Since rad = 180o

=

=

97.42

18075.0rad75.0

Always use = 3.142

(a) 0.6 rad =

[34.37o]

(b) 1.54 rad =

[88.22o]

(c) =rad5

18

[206.24o]

(d) =rad7

32

[139.13o]

(e) 1.2rad =

[216o]

(f) =rad3

[60o]

(g) -0.25 rad =

[-14.32o]

(h) - =rad3

12

[-133.67o]

(i) (1 + ) rad =

[237.29o]

Circular Measure 2

0.75 rad

O

P


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2. ARC LENGTH OF A CIRCLE

2.1 Arc Length, Radius and Angler

S= => S= r

For each of the following, find the length of the arc Sgiven the values ofrand .

Example:

r= 10 cm and = 1.5 rad

Since S= r

= 10 1.5= 15 cm

(a) r= 20 cm and = 0.6 rad

[12 cm]

(b) r= 15 cm and = 1.4 rad

[21 cm]

(c) r= 30 cm and =6

rad

[15.71 cm](d) r= 10 cm and = 60o

[10.47 cm]

(e) r= 1.5 m and = 82.8o

[2.168 m]

For each of the following, calculate the length of the radius rgiven the values ofSand .

Example:

S= 4.5 cm and = 1.5 radSince S= r

4.5 = 1.5r

5.1

5.4=r

= 3 cm

(a) S= 9 cm and = 1.5 rad

[6 cm]

(b) S= 4.5 cm and = 1.8 rad

[2.5 cm]

(c) S= 9 m and = 30o

[17.19 m]

(d) S= 15 cm and = 32.8o

[26.20 cm]

(e) S= 10 cm and =8

rad

[25.46 cm]

Circular Measure 3

O

P

Q

1.5 radS

10cm

O

B

1.5 rad

r 4.5cm


4/9

For each of the following, find the angle in radian given the values ofrand S.

Example:

r= 5 cm and S= 10 cm

Since S= r

10 = 5

=5

10

= 2 rad

(a) r= 9 cm and S= 36 cm

[4 rad]

(b) r= 7 cm and S= 5.6 cm

[0.8 rad]

(c) r= 20 cm and S= 0.7 m

[3.5 rad]

(d) r= 1.2 cm and S= 20 cm

cm]3

216[

(e) r= 3.5 cm and S= 112 cm

[32 cm]

For each of the following, find the angle in degree given the values ofrand S.

Example:r= 8 cm and S= 20 cm

Since S= r

20 = 8

=8

20rad

=

180

5.2

= 143.22o

(a) r= 10 cm and S= 6 cm

[34.37o]

(b) r= 5 cm and S= 5.4 cm

[57.29o]

(c) r= 5.6 cm and S= 12 cm

[122.76o]

(d) r= 7.15 cm and S= 0.28 m

[224.34o]

(e) r= 5.6 cm and S= 3 cm

[96.43o]

Circular Measure 4

O

B

10cm

5cm

O

B

20cm

8cm


5/9

3. AREA OF SECTOR

3.1 Area of Sector, Radius and Angle

Area of a circle,A = r2

Area for a sectorOPQ of angle o,2

360rA

=

Converting to radian,

2

rad2

radr

=

A =2

1r

2, must be in radian.

For each of the following, find the area of the sectorA given the values ofrand .

Example:r= 3 cm and = 1.8 rad

Since 2

2

1rA=

8.132

1 2=

= 8.1 cm2

(a) r= 5 cm and = 0.8 rad

[10 cm2]

(b) r= 2.11 cm and = 1.78 rad

[3.962 cm2]

(c) r= 2.7 cm and =3

21 rad

[6.075 cm2]

(d) r= 12 mm and =5

4rad

[180.98 mm2]

(e) r= 1.5 m and = 122.8o

[2.411 m2]

(f) r=5

3m and = 85o

[0.2671 m2]

(g) r= 13.2 cm and = 67.2o

[102.19 cm2]

Circular Measure 5

O

Q

1.8 rad

3cm

3cm

O

r

r

P

Q


6/9

For each of the following, find the length of the radius rgiven the values ofandA.

Example: = 1.2 rad andA = 15 cm2

Since 2

2

1rA=

15 2.12

1 2= r

2.1

2152 =r

= 25

r= 5 cm

(a) = 3 rad andA = 6 cm2

[2 cm]

(b) = 1.4 rad andA = 118.3 cm2

[13 cm]

(c) =6

rad andA = 3m2

[6 m]

(d) =7

31 rad andA = 78.25 cm2

[109.55 cm]

(e) = 102o andA = 215 cm2

[15.54 cm]

(f) = 65o andA = 78 cm2

[11.73 cm]

(g) = 102o andA = 1215 cm2

[36.94 cm]

Circular Measure 6

O

P Q

r r1.2 rad


7/9

For each of the following, find the angle in radian given the values ofrandA.

Example: r= 5 cm andA =8

39 cm2

Since 2

2

1rA=

= 25

2

1

8

39

258

275

=

4

3= rad

(a) r= 2.4 cm andA = 1.44 cm2

[0.5 rad]

(b) r= 8 cm andA = 64 cm2

[2 rad]

(c) r= 0.6 m andA = 3.24 m2

[18 rad]

For each of the following, find the angle in degree given the values ofrandA.

Example: r= 4 cm andA = 16 cm2

Since 221 rA=

= 242

116

16

216=

= 2 rad

=142.3

1802

= 114.58o

(a) r= 15 cm andA = 50cm

[80o]

(b) r= 8 cm andA = 48 cm2

[85.93o]

(c) r= 3.2 m andA = 5 m2

[55.95o]

Circular Measure 7

O

A

B

16 cm2

4cm

4cm

5 cm

A

BO

cm2938

_5c

m


8/9

2.2 Perimeter of Segments of a Circle

Example:

For the shaded segment in the given diagram beside, find its perimeter.

The perimeter of the shaded segment is the total length of the arc AB

and the chordAB.

SAB = r= 10 1.2

= 12 cm

To find the length of the chordAB, consider

the triangle OAB in the diagram beside.

)142.3

1806.0sin(10

2

1

rad6.0sin10

2

1

=

=

AB

AB

AB = 2 5.646

= 11.292 cmperimeter of shaded segment = 12 + 11.292

= 23.292 cm

3.2 Area of Segments of a Circle

Example:

With reference to the sector in the same diagram beside,

find the area of the shaded segment.

The area of the shaded segment is the difference between

the area of the sectorOAB and the triangle OAB.

Area of sectorOAB =2

1r2

2.1102

1 2=

= 60 cm2

To find the area of the triangle OAB,

please refer to the diagram beside.

ACis perpendicular to the radius OB.

=

=

142.3

1802.1sin

rad2.1sin

AOAC

AO

AC

2cm60.46

9320.010102

1

75.68sin2

1

2

1ofArea

=

=

=

=

OAOB

ACOBOAB

Area of shaded segment = 60 46.60= 13.40 cm2

Circular Measure 8

Note:

The length of the chord

AB =2

sin2

r

Area of segment

( )

sin2

1

sin2

1

2

1

2

22

=

=

r

rr

O

B

10cm

10cm

=

=

0.6rad

0.6rad

O

B

10cm

10cm

1.2 rad

C

Note:

Area of triangle OAB

= sin2

1 2r

O

B

10cm

10cm

1.2 rad

O

B

10cm

10cm

1.2 rad


9/9

For each of the following diagrams, find

(i) the perimeter and

(ii) the area of the shaded segment.

(a)

[Perimeter = 4.974 cm, Area = 0.2579 cm2]

(b)


(c)


(d)


Circular Measure 9

O

P

Q

6c

m6cm

0.8ra

dO

B

5cm

5cm

0.5 rad

O Q

3.8

cm

3.8cm

1 rad

OA

B

4.2cm

4.2

cm

60o

7 Circular Measures

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