This question tests recall of the sector formulae. In each formula, r is the radius of the circle and θ is the angle at the centre, in degrees.
(a)Write down the formula for the arc length of a sector.(1)
(b)Write down the formula for the area of a sector.(1)
(Total for Question 1 is 2 marks)
2
The diagram shows a sector of a circle with centre O. The angle at the centre is 90 degrees.
Diagram NOT accurately drawn
(a)Write the sector's angle as a fraction of 360 degrees. Give your answer in its simplest form.(1)
(b)Write this fraction as a percentage.(1)
(Total for Question 2 is 2 marks)
3
A semicircular rug has a straight edge of length 10 cm and a curved edge. The radius of the semicircle is 5 cm. Calculate the perimeter of the rug, giving your answer correct to 3 significant figures.
Diagram NOT accurately drawn
(Total for Question 3 is 3 marks)
4
A sector of a circle has centre O, radius 4 cm and angle 90 degrees. Find the exact arc length of the sector, giving your answer in terms of π.
Diagram NOT accurately drawn
(Total for Question 4 is 3 marks)
5
A sector of a circle has centre O, radius 9 cm and angle AOB = 60 degrees. Calculate the area of the sector, giving your answer correct to 3 significant figures.
Diagram NOT accurately drawn
(Total for Question 5 is 3 marks)
6
A sector of a circle has centre O, radius 7 cm and angle AOB = 130 degrees. Calculate the arc length of the sector, giving your answer correct to 3 significant figures.
Diagram NOT accurately drawn
(Total for Question 6 is 4 marks)
7
A circular pizza has radius 10 cm. Priya cuts a slice with a sector angle of 72 degrees. Calculate the perimeter of the pizza slice, giving your answer correct to 3 significant figures.
Diagram NOT accurately drawn
(Total for Question 7 is 4 marks)
8
An arc of a circle has length 22 cm. The radius of the circle is 14 cm. Calculate the angle of the sector, giving your answer to the nearest degree.
Diagram NOT accurately drawn
(Total for Question 8 is 4 marks)
9
A garden sprinkler sprays water over a sector of angle 115 degrees. The arc traced by the spray at its maximum reach is 18.4 m long. Calculate the radius of the sprinkler's reach, giving your answer correct to 3 significant figures.
Diagram NOT accurately drawn
(Total for Question 9 is 4 marks)
10
A sector of a circle has area 154 cm2 and angle 100 degrees. Calculate the radius of the circle, giving your answer correct to 3 significant figures.
Diagram NOT accurately drawn
(Total for Question 10 is 5 marks)
11
A sector of a circle has area 60 cm2 and radius 11 cm. Calculate the angle of the sector, giving your answer to the nearest degree.
Diagram NOT accurately drawn
(Total for Question 11 is 4 marks)
12
The minute hand of a clock is 8 cm long. Calculate the area swept out by the minute hand in 25 minutes, giving your answer correct to 3 significant figures.
Diagram NOT accurately drawn
(Total for Question 12 is 5 marks)
13
Two adjacent slices are cut from a circular pizza of radius 15 cm, meeting along a shared straight edge from the centre. Aaliyah's slice has a sector angle of 50 degrees and Tomasz's slice, next to it, has a sector angle of 80 degrees. Calculate the difference between the areas of the two slices, giving your answer correct to 3 significant figures.
Diagram NOT accurately drawn
(Total for Question 13 is 5 marks)
14
A circular flower bed has radius 6 m. A circular path of radius 2 m is removed from the centre, leaving a ring-shaped bed. A gardener plants flowers in the part of the ring between two markers that are 150 degrees apart, measured from the centre. Calculate the area of the planted region, giving your answer correct to 3 significant figures.
Diagram NOT accurately drawn
(Total for Question 14 is 5 marks)
15
A car's windscreen wiper blade sweeps through an angle of 100 degrees. The tip of the blade is 32 cm from the pivot, and the blade only cleans the glass from 5 cm out from the pivot to the tip (it does not clean the small circle close to the pivot). Calculate the area of glass cleaned by the wiper, giving your answer correct to 3 significant figures.
Diagram NOT accurately drawn
(Total for Question 15 is 6 marks)
16
A sector of a circle has centre O, radius 12 cm and angle AOB = 80 degrees. The chord AB divides the sector into a triangle OAB and a shaded segment, as shown.
Diagram NOT accurately drawn
(a)Calculate the area of the segment, giving your answer correct to 3 significant figures.(3)
(b)Calculate the perimeter of the segment, giving your answer correct to 3 significant figures.(3)
(Total for Question 16 is 6 marks)
17
A sector of a circle has radius 9 cm. The arc length of the sector is exactly 6pi cm.
Diagram NOT accurately drawn
(a)Find the angle of the sector.(2)
(b)Hence find the exact area of the sector, giving your answer in terms of π.(3)
(Total for Question 17 is 5 marks)
18
A sector of a circle has radius (x + 3) cm and angle 40 degrees. The arc length of the sector is 9.2 cm. Form an equation in x and solve it to find the value of x, giving your answer correct to 3 significant figures.
Diagram NOT accurately drawn
(Total for Question 18 is 4 marks)
19
Sector A has radius 5 cm and area 12 cm2. Sector B has the same angle as sector A, but has radius 15 cm. Find the area of sector B. You do not need to find the angle.
Diagram NOT accurately drawn
(Total for Question 19 is 4 marks)
20
A sector-shaped garden ornament is cut from a sheet of metal. The sector has radius 18 cm. Its perimeter (the two straight edges plus the curved arc) is 54 cm. The metal costs 0.09 pounds per cm2.
Diagram NOT accurately drawn
(a)Show that the arc length of the sector is 18 cm.(2)
(b)Find the angle of the sector, giving your answer correct to 1 decimal place.(2)
(c)Find the area of the sector, giving your answer correct to 3 significant figures.(2)
(d)Calculate the cost of the metal needed for the ornament.(1)
(Total for Question 20 is 7 marks)
Mark scheme · 5.17D Sector Areas and Arc Lengths: Fluency and Exam Drill
Question 1
(a) B1 arc length = (θ/360) x 2 x π x r oe, e.g. (θ/360) x π x d
(a) Answer: Arc length = (θ / 360) x 2 x π x r
(b) B1 area = (θ/360) x π x r2 oe
(b) Answer: Area = (θ / 360) x π x r2
Question 2
(a) B1 1/4 oe, must be simplified
(a) Answer: 1/4
(b) B1 25% ft their fraction from part (a)
(b) Answer: 25%
Question 3
M1 arc length = π x 5 oe, or (180/360) x 2 x π x 5
A1 arc length = 15.7 cm awrt (15.70796...)
A1 perimeter = their arc + 10 = 25.7 cm awrt, cao
Answer: 25.7 cm (3 s.f.)
Question 4
B1 identifies fraction 90/360 = 1/4 oe
M1 (1/4) x 2 x π x 4 oe
A1 2pi cm cao (exact; do not accept a decimal-only answer)
Answer: 2pi cm (= 6.28 cm to 3 s.f.)
Question 5
M1 (60/360) x π x 92 oe
A1 13.5pi or 42.41... seen
A1 42.4 cm2 awrt, cao
Answer: 42.4 cm2 (3 s.f.)
Question 6
M1 full circumference = 2 x π x 7 (= 43.98...) oe
M1 (130/360) x 2 x π x 7 oe
A1 15.88... seen (awrt 15.9)
A1 15.9 cm awrt, cao
Answer: 15.9 cm (3 s.f.)
Question 7
M1 arc length = (72/360) x 2 x π x 10 oe
A1 arc length = 12.6 cm awrt (12.566)
M1 perimeter = their arc length + 2 x 10 oe
A1 32.6 cm awrt, cao
Answer: 32.6 cm (3 s.f.)
Question 8
M1 sets up 22 = (θ/360) x 2 x π x 14 oe
M1 rearranges to θ = 22 x 360 / (2 x π x 14) oe
A1 90.0... seen (awrt 90.0)
A1 90 degrees cao, to the nearest degree
Answer: 90 degrees (nearest degree)
Question 9
M1 sets up 18.4 = (115/360) x 2 x π x r oe
M1 rearranges to r = 18.4 x 360 / (2 x π x 115) oe
A1 9.16... or 9.17 seen
A1 9.17 m awrt, cao
Answer: 9.17 m (3 s.f.)
Question 10
M1 sets up 154 = (100/360) x π x r2 oe
M1 rearranges to r2 = 154 x 360 / (π x 100) oe
A1 r2 = 176 (awrt) seen
M1 square roots their r2 value
A1 13.3 cm awrt, cao
Answer: 13.3 cm (3 s.f.)
Question 11
M1 sets up 60 = (θ/360) x π x 112 oe
M1 rearranges to θ = 60 x 360 / (π x 121) oe
A1 56.8... seen (awrt 56.8)
A1 57 degrees cao, to the nearest degree
Answer: 57 degrees (nearest degree)
Question 12
M1 25 minutes is 25/60 of a full turn oe
A1 angle = 150 degrees
M1 (150/360) x π x 82 oe
A1 83.7... seen (awrt 83.8)
A1 83.8 cm2 awrt, cao
Answer: 83.8 cm2 (3 s.f.)
Question 13
M1 area of 50 degree slice = (50/360) x π x 152 oe
M1 area of 80 degree slice = (80/360) x π x 152 oe
M1 valid method to find the difference between the two areas (may use the 30 degree angle difference directly) oe
A1 58.9... seen (awrt 58.9)
A1 58.9 cm2 awrt, cao
Answer: 58.9 cm2 (3 s.f.)
Question 14
M1 area of large sector = (150/360) x π x 62 oe
M1 area of small sector = (150/360) x π x 22 oe
M1 subtracts small sector area from large sector area oe, or uses (150/360) x π x (62 - 22)
A1 41.8... seen (awrt 41.9)
A1 41.9 m2 awrt, cao
Answer: 41.9 m2 (3 s.f.)
Question 15
M1 area of outer sector (radius 32 cm) = (100/360) x π x 322 oe
A1 893.6... cm2 for the outer sector seen (awrt 894)
M1 area of inner sector (radius 5 cm) = (100/360) x π x 52 oe
A1 21.8... cm2 for the inner sector seen (awrt 21.8)
M1 subtracts inner sector area from outer sector area oe
A1 872 cm2 awrt, cao
Answer: 872 cm2 (3 s.f.)
Question 16
(a) M1 area of sector = (80/360) x π x 122 oe (= 100.5)
(a) M1 area of triangle OAB = 0.5 x 122 x sin(80) oe (= 70.9)
(a) A1 29.6 cm2 awrt, cao (sector area minus triangle area)
(a) Answer: 29.6 cm2 (3 s.f.)
(b) M1 arc length = (80/360) x 2 x π x 12 oe (= 16.8)
(b) M1 chord AB = 2 x 12 x sin(40) oe (= 15.4)
(b) A1 32.2 cm awrt, cao (arc length plus chord length)
(b) Answer: 32.2 cm (3 s.f.)
Question 17
(a) M1 sets up 6pi = (θ/360) x 2 x π x 9 oe, i.e. 6pi = (θ/360) x 18pi
(a) A1 θ = 120 degrees cao
(a) Answer: 120 degrees
(b) M1 identifies fraction 120/360 = 1/3 oe
(b) M1 (1/3) x π x 92 oe
(b) A1 27pi cm2 cao (exact; do not accept a decimal-only answer)
(b) Answer: 27pi cm2 (= 84.8 cm2 to 3 s.f.)
Question 18
M1 sets up 9.2 = (40/360) x 2 x π x (x + 3) oe
M1 rearranges to (x + 3) = 9.2 x 360 / (2 x π x 40) oe
A1 x + 3 = 13.1... seen (awrt 13.2)
A1 x = 10.2 awrt, cao
Answer: x = 10.2 (3 s.f.)
Question 19
M1 recognises that sectors with equal angle are similar, so areas scale with the square of the radius ratio oe
M1 scale factor for length = 15/5 = 3, so area scale factor = 32 = 9 oe
M1 12 x 9 oe
A1 108 cm2 cao
Answer: 108 cm2
Question 20
(a) M1 perimeter = 2 x 18 + arc length, i.e. 54 = 36 + arc length oe