circular measure na

7/29/2019 Circular Measure NA

1/26

1. Find the

(a) arc length

(b) area of the sector of circle with centre O.

(a) ________________________[1]

225


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(b) ________________________[1]

2. The diagram below shows a sector AOB of radius 8 cm and arc length 12 cm.

Calculate

(a) AOB in terms of degrees

(b) the length AB

(c) the perimeter of the shaded region

BA

O


3/26

(a) ________________________[2]

(b) ________________________[3]

(c) ________________________[1]

3a) Convert 160 to radians, giving your answer to one decimal place. [1]

3b) The diagram shows a cross section of a drainage pipe of radius 36 cm. The drainage pipe contains some

water.

i) Find the area of the triangle ORS. [1]

ii) Find the area of the segment occupied by the water. [2]

R S

O

110 36cm

water


4/26

4. The shaded area in the diagram represents the part of the flat windscreen of a car which is being wiped

by the windscreen wiperAB.

The wiper rotates through 150 about O.

OA = OA = 7 cm andAB =AB= 35 cm.

(a) Leaving your answer in terms of,

calculate

(i) the length of the arc AA, [2]

(ii) the length of the arcBB, [2]

(iii) the ratio of the arc lengths AA:BB [1]

(b) Taking the value of 3.142= ,

calculate

(i) the area of the screen which is wiped, [3]

(ii) the perimeter of the screen which is wiped. [2]


5/26

5. The diagram below shows a sectorAOB of radius 7 cm. Find

(a) the length of arcAB, [2]

(b) the area of sectorAOB, [2]

(c) the area of triangleAOB, [2]

(d) the area of the shaded segment. [2]

0.7 rad

O

B

A7 cm

7 cm


6/26

6. The diagram shows the layout of a childrens playground.PQR is a triangle with QR = 22 m and PR = 25

m.PRSis a sector of a circle, centreR, radius 25 m. QRSis a straight line.

Calculate(a) Calculate the length ofPQ. [2]

(b) A slide is fixed at R. The angle of elevation from Q to the top of the slide is

7.8. Calculate the height of the slide. [3]

(c) Bushes are planted along the curved boundary,PS, of the playground.Calculate the perimeter of the playground which is surrounded by the bushes. [2]

P

Q

R

S

25 m

22 m

110


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The diagram shows a major segment of a circle, centre O, radius 12 cm.A,B andXare

points on the circumference.AngleAOB = 150, chordAB = 23 cm.

Calculate(a) the arc length of the major sectorOAXB,

(b) the perimeter of the major segment.

Give your answers to both questions in terms of.

A

O

B

X

1223

150


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Answer (a) cm. [2]

(b) cm. [1]


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8. The diagram below is a sectorOPQ, centre O, ando120=POQ .

(a) Given that the arc length is 2

12 cm, find the radius.

(b) Find the sector area, leaving your answer in .

Answer (a) ___________________________ cm

(b)__________________________ cm2

[2]

[2]

O

P

Q

120o


10/26

9

.

OXYis a sector of a circle with the centerO and radius 12 cm. AngleXOY=

1.25 radians. Calculate the area of the shaded segment. [3]

1.25

12 cm

X Y

O


11/26

10. In the diagram,POQ is a sector of a circle with centre O and radius 10 cm.

Given that QR is perpendicular to OPandPR= 4 cm, find

(i) POQ in radians, [2]

(ii) the area of the shaded region. [3]


12/26

11.

The figure shows part of the circle with centre O and radius 10 cm.

Given that radPOQ 2.1= , find

(a) the perimeter of the shaded region, [3]

(b) the area of the shaded region. [3]

10 cm

1.2 rad

10 cm

Q

O

P

R


13/26

12. The diagram shows the sectorOAB of a circle, centre O, and radius 8 cm. It is given thatthe length of arcAB is 10 cm.

(a) Show that angleAOB is 1.25 radian. [1]

(b) Find the area of the shaded region.


14/26

Answer : (b) ________________ cm2 [2]

13. The diagram below shows a plot of land in the shape of sectorOABCD with centre O,

radius 3.5m.

ArcsAB = BC = CD. It is given that 6

5=AOD andPQ = 0.8448m.

i) Show that OP = 0.9995m. [3]

ii) Find the area of the shaded region. [3]

O

A

B C

DP Q

6

5

0.8448

m

3.5m


15/26

14. The diagram shows a right-angled triangle OPQ. An arc of a circle with

centre O and radius OPcuts OQ atR. Given POQ = 0.84 radians,

OP= 6 cm andPQ = 6.7 cm, calculate

(a) the area of sectorOPR,

(b) the area of the shaded region,

(c) the perimeter of the shaded region.


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Answer: (a) [2]

(b) [2]

(c) [2

15. A,B and Care three points on the circumference of a circle with centre O.

AB is a diameter of the circle and , angleAOC= x o.

(a) Giving your reasons, find, in terms of x ,

(i) angleABC, [2]

(ii) reflex angle AOC. [2]

(b) Given that that length of the radius OB is 3.5 cm and obtuse angle BOCis 2.4 radians, calculate

(i) the area of sectorBOC . [2]

(ii) the area of triangle BOC . [2]

.

x


17/26

16

In the diagram,POQ is a sector of a circle with centre O and radius 10 cm. Given that

QR is perpendicular to OPandPR= 6 cm, find

(a) ROQ in radians,(b) the area of the shaded region.

O

10 cm


18/26

Answer: (a)....[3]

(b) ...cm2[3]

17. The diagram, which shows the sectorAOB of a circle, represents a piece of

card. The radius of the sector is 21 cm and the angle AOB is 120.

(a) Calculate

(i) the length of the arcACB, [2]

(ii) the area of the sector AOB, [2]

(iii) the area of the segment ACB. [2]

(b) The card is used to make a hollow cone by joining the edges OA

and OB. Calculate

(i) the radius of the base, [1]

(ii) the height of the cone. [1]


19/26

18. In the diagram, OPQ is a sector of a circle, centre O and radius 15 cm. OPis parallel to SR.The length of the arcPQ is 18 cm and Sis the mid-point ofOQ.

Find(i) POQ in radians, [1]

(ii) the length ofRT, [2]

(iii) ROT in radians, [2]

(iv) the area of the shaded region. [3]


20/26

19. A circle, centre O, has radius 7 cm.

Pand Q are points on the circumference such that anglePOQ = 1.4 radians.

Calculate

(a) the perimeter of the minor sectorPOQ,

(b) the area of the major sectorPOQ.

7 cm

Q

P

O

1.4 rad


21/26

Answer: (a)______________cm [2]

(b)_____________cm2 [2]

21 The diagram below shows a right-angled triangle OPQ. An arc of a circle with centre O

and radius OPcuts OQ atR.

Given that POQ = 0.84 radian, OP= 6 cm andPQ = 6.7 cm, calculate

(a) the area of sectorOPR, [1]

(b) the area of the shaded region, [1](c) the perimeter of the shaded region. [2]

O

P

R Q

6 cm 6.7 cm

0.84 rad


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22.

A,B, CandD are points on a circle with centre 0 and radius 8 cm.

Given that = 100AOC

Calculate

(a) ADC [1]

(b) ABC [1]

(c) the length of the chord AC . [2]

(d) area of OAC [2]

(e) area of segment ABC . [2]

D

C

AO8 cm

B

100 8 cm


23/26

O

A B

12

3

2

or

miners

se

23. The pointsA andB lie on a circle, centre O.

OA = OB = 12 cm and BOA =3

2radians.

Calculate(a) the length of the arcAB,(b) the area of the sector OAB,

(c) the area of the shaded segment.


24/26

Answer (a) ........................................... cm [2]

(b) ......................................... cm2 [2]

(c) ......................................... cm2 [3]

24

.Figure 1 shows a circle, centre O,

passes through A, B, C and D.

Given that the radius of the circle is 9 cm

andDOC = 108o.

Calculate

(i) DBC, [1]

(ii) ABD, [2]

(iii) area of minor sector DOC, [2]

(iv) area of the segment DEC. [2]

B

C

Figure 1

O

A

D108o

E


25/26

10 cm

25 The diagram shows the sector of a circle.

The radius of the circle is 10 cm.

The angle AOB is 2.51 radians.A

O

B

2.51 rad


26/26

h10 cm

r

O

(a) Find the length of the arc AB. [2]

(b) Find the area of the sector AOB. [1]

(c) OA is joined to OB to make a cone as shown.

(i) Show that the radius, r, of the cone is 6 cm. [1]

(ii) Find the height, h, of the cone. [2](iii) Find the volume of the cone. [2]

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