The diagram shows a circle with centre O. Points A and B lie on the circumference. Radii OA and OB are drawn, and the minor arc AB is marked. The region enclosed by OA, OB and the arc AB is shaded.
A) Segment
B) Sector
C) Chord
D) Semicircle
(Total for Question 1 is 1 mark)
2
A sector of a circle has centre O and radius 8 cm. The angle at the centre, angle AOB, is 90 degrees.
Diagram NOT accurately drawn
(Total for Question 2 is 2 marks)
3
A sector of a circle has centre O and radius 10 cm. The angle at the centre is 90 degrees.
Diagram NOT accurately drawn
(Total for Question 3 is 2 marks)
4
A semicircular lawn has a diameter of 12 m.
Diagram NOT accurately drawn
(Total for Question 4 is 3 marks)
5
A sector of a circle has centre O, radius 6 cm and angle AOB = 60 degrees.
Diagram NOT accurately drawn
(Total for Question 5 is 2 marks)
6
A sector of a circle has centre O, radius 9 cm and angle AOB = 150 degrees.
Diagram NOT accurately drawn
(Total for Question 6 is 3 marks)
7
A sector of a circle has centre O, radius 7 cm and angle AOB = 120 degrees. Calculate the perimeter of the sector OAB, giving your answer correct to 3 significant figures.
Diagram NOT accurately drawn
(Total for Question 7 is 3 marks)
8
A sector of a circle has centre O, radius 12 cm and angle AOB = 150 degrees. Find the exact length of the arc AB, giving your answer in terms of π.
Diagram NOT accurately drawn
(Total for Question 8 is 2 marks)
9
A sector of a circle has centre O, radius 9 cm and angle AOB = 40 degrees. Find the exact area of the sector, giving your answer in terms of π.
Diagram NOT accurately drawn
(Total for Question 9 is 3 marks)
10
A circular pizza has a diameter of 32 cm. Fatima cuts a slice which forms a sector with an angle of 45 degrees at the centre.
Diagram NOT accurately drawn
(a)Calculate the length of the curved crust edge of the slice, giving your answer correct to 3 significant figures.(2)
(b)Calculate the area of the slice, giving your answer correct to 3 significant figures.(2)
(Total for Question 10 is 4 marks)
11
A sector of a circle has centre O and radius 10 cm. The length of the arc AB is 15 cm.
Diagram NOT accurately drawn
(Total for Question 11 is 3 marks)
12
A sector of a circle has an angle at the centre of 72 degrees and an area of 40 cm2.
Diagram NOT accurately drawn
(Total for Question 12 is 3 marks)
13
The diagram shows a rectangular garden ABCD, where AB = 12 m and BC = 8 m. A quarter-circle flower bed, with centre A and radius 4 m, is cut from the corner of the garden at A.
Diagram NOT accurately drawn
(a)Calculate the area of the flower bed, giving your answer correct to 3 significant figures.(2)
(b)Calculate the area of the remaining garden (the rectangle with the flower bed removed), giving your answer correct to 3 significant figures.(3)
(Total for Question 13 is 5 marks)
14
A sector of a circle has radius r cm and angle 80 degrees at the centre. The perimeter of the sector is 30 cm.
Diagram NOT accurately drawn
(Total for Question 14 is 3 marks)
15
The diagram shows a shape made from two sectors with the same centre O and the same angle of 100 degrees. The outer radius is 15 cm and the inner radius is 9 cm.
Diagram NOT accurately drawn
(Total for Question 15 is 4 marks)
16
The diagram shows a sector OAB with centre O, radius 8 cm and angle AOB = 90 degrees. The chord AB divides the sector into a triangle OAB and a shaded segment.
Diagram NOT accurately drawn
(Total for Question 16 is 4 marks)
17
A sector of a circle has radius 20 cm. It is bent round and its straight edges are joined to form the curved surface of a cone with base radius 8 cm. The radius of the sector becomes the slant height of the cone, and the arc length of the sector becomes the circumference of the base of the cone.
Diagram NOT accurately drawn
(Total for Question 17 is 4 marks)
18
A school running track is made from two straight sections and two semicircular ends. Each straight section is 80 m long. Each semicircular end has a radius of 32 m.
Diagram NOT accurately drawn
(a)Calculate the total distance around one complete lap of the track, giving your answer correct to 3 significant figures.(3)
(b)Anwar runs 5 complete laps of the track. Calculate the total distance he runs, in kilometres, giving your answer correct to 3 significant figures.(3)
(Total for Question 18 is 6 marks)
19
A sector of a circle has centre O. The area of the sector is 54pi cm2. The length of the arc of the sector is 12pi cm.
Diagram NOT accurately drawn
(a)Show that the radius of the sector is 9 cm.(3)
(b)Calculate the angle of the sector.(2)
(Total for Question 19 is 5 marks)
20
A security light has a motion sensor that detects movement up to 12 m away, within an angle of 110 degrees, as shown in the diagram.
Diagram NOT accurately drawn
(a)Calculate the area of the region covered by the sensor, giving your answer correct to 3 significant figures.(3)
(b)The homeowner wants the sensor to cover an area of at least 150 m2, without increasing the 12 m range. State, giving a reason, whether an angle of 110 degrees is sufficient.(3)
(Total for Question 20 is 6 marks)
Mark scheme · 5.17 Sector Areas and Arc Lengths
Question 1
B1 B (Sector)
Answer: B
Question 2
M1 (90/360) x 2 x π x 8 oe, or 2 x π x 8 = 50.2... seen
A1 awrt 12.6 (cm)
Answer: 12.6 cm
Question 3
M1 (90/360) x π x 102 oe, or π x 102 = 314... seen
A1 awrt 78.5 (cm2)
Answer: 78.5 cm2
Question 4
M1 π x 6 oe (half the circumference of a circle of radius 6 m)
M1 their arc length + 12 (adds the diameter to complete the perimeter)
A1 awrt 30.8 (m)
Answer: 30.8 m
Question 5
M1 (60/360) x 2 x π x 6 oe
A1 awrt 6.28 (cm)
Answer: 6.28 cm
Question 6
M1 (150/360) x π x 92 oe
M1 (150/360) x 254.469... (correct substitution seen)
A1 awrt 106 (cm2)
Answer: 106 cm2
Question 7
M1 (120/360) x 2 x π x 7 oe (arc length AB)
M1 their arc length + (2 x 7) (adds the two radii OA and OB)
A1 awrt 28.7 (cm)
Answer: 28.7 cm
Question 8
M1 (150/360) x 2 x π x 12 oe, or 24pi seen
A1 10pi (cm) cao (accept exact equivalent forms, e.g. 10 x π)
Answer: 10pi cm
Question 9
M1 (40/360) x π x 92 oe, or 81pi seen
M1 (1/9) x 81pi (correct simplification of the fraction of the circle)
A1 9pi (cm2) cao
Answer: 9pi cm2
Question 10
(a) M1 (45/360) x 2 x π x 16 oe (radius = 16 cm identified)
(a) A1 awrt 12.6 (cm)
(a) Answer: 12.6 cm
(b) M1 (45/360) x π x 162 oe
(b) A1 awrt 101 (cm2)
(b) Answer: 101 cm2
Question 11
M1 2 x π x 10 (= 62.8...) (full circumference found)
M1 (15 / their circumference) x 360 oe (correct proportion set up)
A1 awrt 85.9 (degrees)
Answer: 85.9 degrees
Question 12
M1 40 = (72/360) x π x r2 oe (correct equation formed)
M1 r2 = 40 x 360 / (72 x π) (= 63.6...) (correct rearrangement)
A1 awrt 7.98 (cm)
Answer: 7.98 cm
Question 13
(a) M1 (90/360) x π x 42 oe
(a) A1 awrt 12.6 (m2)
(a) Answer: 12.6 m2
(b) M1 12 x 8 (= 96) (area of rectangle found)
(b) M1 96 - their (a) (correct subtraction of the flower bed area)
(b) A1 awrt 83.4 (m2) ft their (a)
(b) Answer: 83.4 m2
Question 14
M1 2r + (80/360) x 2 x π x r = 30 oe (correct equation formed)
M1 r(2 + (80/360) x 2 x π) = 30, or r x 3.396... = 30 (correct factorisation/simplification)
A1 awrt 8.83 (cm)
Answer: 8.83 cm
Question 15
M1 (100/360) x π x 152 and (100/360) x π x 92 (both sector areas found), or (100/360) x π x (152 - 92) oe
M1 152 - 92 = 144 (correct difference of the two radii squared)
M1 (100/360) x π x 144 (= 40pi) (correct substitution)
A1 awrt 126 (cm2)
Answer: 126 cm2
Question 16
M1 (90/360) x π x 82 (= 16pi) (sector area found)
M1 0.5 x 8 x 8 (= 32) (area of right-angled triangle OAB found)
M1 their sector area - their triangle area (correct subtraction)
A1 awrt 18.3 (cm2)
Answer: 18.3 cm2
Question 17
M1 2 x π x 8 (= 16pi) (base circumference found)
M1 (θ/360) x 2 x π x 20 = 2 x π x 8 oe (correct equation formed, equating arc length to base circumference)
M1 θ = (8/20) x 360 oe (correct rearrangement)
A1 144 (degrees) cao
Answer: 144 degrees
Question 18
(a) M1 2 x π x 32 oe (recognising the two semicircular ends combine to make one full circle)
(a) M1 their circle circumference + (2 x 80) (adds the two straight sections)
(a) A1 awrt 361 (m)
(a) Answer: 361 m
(b) M1 their (a) x 5 (= 1805...) (correct multiplication by 5 laps)
(b) M1 their distance in m / 1000 (correct conversion to km)
(b) A1 awrt 1.81 (km) ft their (a)
(b) Answer: 1.81 km
Question 19
(a) M1 54pi = (θ/360) x π x r2 and 12pi = (θ/360) x 2 x π x r (both correct equations formed, or equivalent using Area = 0.5 x r x arc length)
(a) M1 dividing to eliminate θ, e.g. 54pi / 12pi = r/2 oe (correct method to eliminate θ/360)
(a) A1 r = 9 (cm) cso -- fully correct working shown leading to the given answer
(a) Answer: r = 9 cm (shown)
(b) M1 12pi = (θ/360) x 2 x π x 9 oe (correct substitution of r = 9 into the arc length formula)
(b) A1 240 (degrees) cao
(b) Answer: 240 degrees
Question 20
(a) M1 (110/360) x π x 122 oe
(a) M1 (110/360) x 452.389... (correct substitution seen)
(a) A1 awrt 138 (m2)
(a) Answer: 138 m2
(b) B1 correct comparison made using their (a), e.g. 138 (m2) < 150 (m2) ft their (a)
(b) B1 correct conclusion: No, 110 degrees is not sufficient
(b) B1 valid reason given, e.g. area is directly proportional to angle for a fixed radius, so a larger angle than 110 degrees would be needed to reach 150 m2
(b) Answer: No, 110 degrees is not sufficient, because the covered area (138 m2 to 3 sf) is less than 150 m2.