Arcs and Sectors Question Bank
Arcs and Sectors Question Bank
Arcs and Sectors Question Bank
The points A, B, C, D, E and F are on the circle, and [AF] is a diameter. The length of arc ABC
is 6 cm.
2
The area of sector OCDE is 45 cm .
The radius of the smaller circle is 8 cm and the radius of the larger circle is 10 cm.
π
Points A, B and C are on the circumference of the larger circle such that AÔB is radians.
3
3.) The circle shown has centre O and radius 3.9 cm.
Points A and B lie on the circle and angle AOB is 1.8 radians.
4.) The following diagram shows a circle with centre O and radius 4 cm.
diagram not to scale
The points A, B and C lie on the circle. The point D is outside the circle, on (OC).
Angle ADC = 0.3 radians and angle AOC = 0.8 radians.
5.) The following diagram shows a semicircle centre O, diameter [AB], with radius 2.
(b) Explain why the area of triangle OPA is the same as the area triangle OPB.
(3)
Let S be the total area of the two segments shaded in the diagram below.
(d) Find the value of qwhen S is a local minimum, justifying that it is a minimum.
(8)
6.) The diagram below shows a circle centre O, with radius r. The length of arc ABC is 3π cm and AÔC
2π
= .
9
7.) The following diagram shows a sector of a circle of radius r cm, and angle qat the centre. The
perimeter of the sector is 20 cm.
20 2r
(a) Show that q= .
r
2
(b) The area of the sector is 25 cm . Find the value of r.
(Total 6 marks)
8.) The following diagram shows the triangle AOP, where OP = 2 cm, AP = 4 cm and AO = 3 cm.
A
diagram not to
scale
O P
The following diagram shows two circles which intersect at the points A and B. The smaller
circle C1 has centre O and radius 3 cm, the larger circle C2 has centre P and radius 4 cm, and OP
= 2 cm. The point D lies on the circumference of C1 and E on the circumference of C2.Triangle
AOP is the same as triangle AOP in the diagram above.
A
C2
C1
O P diagram not to
D E
scale
2
(d) The area of the quadrilateral AOBP is 5.81 cm .
4 2
The area of sector OABC is π and the length of arc ABC is π.
3 3
10.) The diagram below shows a circle of radius r and centre O. The angle AÔB = q.
2
The length of the arc AB is 24 cm. The area of the sector OAB is 180 cm .
11.) The following diagram shows two semi-circles. The larger one has centre O and radius 4 cm. The
smaller one has centre P, radius 3 cm, and passes through O. The line (OP) meets the larger semi-circle at S.
The semi-circles intersect at Q.
80
(iii) Hence show that sin OP̂Q = .
9
(c) Consider the larger semi-circle, with centre O. Calculate the area of the sector QOS.
(3)
12.) The following diagram shows a circle of centre O, and radius r. The shaded sector OACB has an area
2
of 27 cm . Angle AÔB = θ = 1.5 radians.
A
C
B
r
O
Working:
Answers:
(a) ..................................................................
(b) ..................................................................
(Total 6 marks)
13.) The diagram below shows a circle of radius 5 cm with centre O. Points A and B are on the circle, and
AÔB is 0.8 radians. The point N is on [OB] such that [AN] is perpendicular to [OB].
5 cm
0.8
O B
N
Working:
Answer:
…………………………………………........
(Total 6 marks)
14.) The diagram below shows a triangle and two arcs of circles.
The triangle ABC is a right-angled isosceles triangle, with AB = AC = 2. The point P is the
midpoint of [BC].
E
B
P
2
A 2 C
Working:
Answers:
(a) ..................................................................
(b) ..................................................................
(Total 6 marks)
15.) The diagram below shows a circle, centre O, with a radius 12 cm. The chord AB subtends at an angle
of 75° at the centre. The tangents to the circle at A and at B meet at P.
A
12 cm
P diagram not to
O 75º scale
(a) Using the cosine rule, show that the length of AB is 12 21 – cos 75 .
(2)
16.) The following diagram shows a circle of centre O, and radius 15 cm. The arc ACB subtends an angle
of 2 radians at the centre O.
C
A B
15
2 rad
O AÔB = 2 radians
OA = 15 cm
Find
Answers:
(a) ..................................................................
(b) ..................................................................
(Total 6 marks)
17.) In the following diagram, O is the centre of the circle and (AT) is the tangent to the circle at T.
O A
If OA = 12 cm, and the circle has a radius of 6 cm, find the area of the shaded region.
Working:
Answer:
.......................................................................
(Total 4 marks)
18.) The diagram below shows a sector AOB of a circle of radius 15 cm and centre O. The angle qat the
centre of the circle is 2 radians.
Working:
Answers:
(a) ..................................................................
(b) ..................................................................
(Total 4 marks)
1 radian
Answer:
......................................................................
(Total 4 marks)
20.) O is the centre of the circle which has a radius of 5.4 cm.
A B
2
The area of the shaded sector OAB is 21.6 cm . Find the length of the minor arc AB.
Working:
Answer:
......................................................................
(Total 4 marks)
21.) The diagrams show a circular sector of radius 10 cm and angle θ radians which is formed into a cone
of slant height 10 cm. The vertical height h of the cone is equal to the radius r of its base. Find the angle θ
radians.
10cm
10cm
h
Working:
Answer:
......................................................................
(Total 4 marks)