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Circle (with Radians)
Question Paper 1
Level
A Level
Subject
Maths
Exam Board
OCR
Module
Core 2
Topic
Trigonometry
Sub Topic
Circle Problems (with Radians)
Booklet
Question Paper - 1
56 minutes
Time Allowed:
Score:
/46
Percentage:
/100
Grade Boundaries:
A*
>85%
A
77.5%
B
C
D
E
U
70%
62.5%
57.5%
45%
<45%
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1
A
C
5.2 cm
1.2 rad
O
2.6 cm
B
D
8 cm
The diagram shows a sector AOB of a circle with centre O and radius 8 cm. The angle AOB is 1.2 radians.
The points C and D lie on OA and OB respectively such that OC = 5.2 cm and OD = 2.6 cm. CD is a straight
line.
(i) Find the area of the shaded region ACDB.
[4]
(ii) Find the perimeter of the shaded region ACDB.
[5]
2
A
B
2
3r
rad
12 cm
O
The diagram shows a sector OAB of a circle, centre O and radius 12 cm. The angle AOB is 23 r radians.
(i) Find the exact length of the arc AB.
[2]
(ii) Find the exact area of the shaded segment enclosed by the arc AB and the chord AB.
[5]
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3
A
B
r cm
72°
O
The diagram shows a sector AOB of a circle, centre O and radius r cm. Angle AOB is 72°.
(i) Express 72° exactly in radians, simplifying your answer.
[1]
The area of the sector AOB is 45π cm2.
(ii) Find the value of r.
[2]
(iii) Find the area of the segment bounded by the arc AB and the chord AB, giving your answer correct to
3 significant figures.
[3]
4
A
B
8 cm
O
The diagram shows a sector AOB of a circle, centre O and radius 8 cm. The perimeter of the sector
is 23.2 cm.
(i) Find angle AOB in radians.
[3]
(ii) Find the area of the sector.
[2]
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5
E
D
11cm
8 cm
A
65°
B
C
The diagram shows two congruent triangles, BCD and BAE , where ABC is a straight line. In
triangle BCD, BD = 8 cm, CD = 11 cm and angle CBD = 65◦ . The points E and D are joined by an
arc of a circle with centre B and radius 8 cm.
(i) Find angle BCD.
[2]
(ii) (a) Show that angle EBD is 0.873 radians, correct to 3 significant figures.
[2]
(b) Hence find the area of the shaded segment bounded by the chord ED and the arc ED, giving
your answer correct to 3 significant figures.
[4]
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6
A
B
O
Fig. 1
Fig. 1 shows a sector AOB of a circle, centre O and radius OA. The angle AOB is 1.2 radians and the
area of the sector is 60 cm2 .
(i) Find the perimeter of the sector.
[4]
A pattern on a T-shirt, the start of which is shown in Fig. 2, consists of a sequence of similar sectors.
The first sector in the pattern is sector AOB from Fig. 1, and the area of each successive sector is 35 of
the area of the previous one.
Fig. 2
(ii) (a) Find the area of the fifth sector in the pattern.
(b) Find the total area of the first ten sectors in the pattern.
[2]
[2]
(c) Explain why the total area will never exceed a certain limit, no matter how many sectors are
used, and state the value of this limit.
[3]