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7/29/2019 Circular Measure NA
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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
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(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
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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]
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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
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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
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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
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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]
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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
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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.
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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
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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
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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
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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]
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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]
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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
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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
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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.
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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
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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
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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]