Circles, arcs and sectors — Paper 1 · IGCSE Mathematics 0580

22 NOT TO SCALE A O B The diagram shows half of a circle, centre O. (a) What is the special name of the line AB? Answer(a) [1] (b) AB = 12 cm. (i) Calculate the perimeter of the shape. Answer(b)(i) cm [2] (ii) Calculate the area of the shape. Answer(b)(ii) cm2 [2]

5 marks

Mark scheme: 22 (a) diameter 1 (b)(i) rounds to 30.8 or 30.9 2 M1 for 0.5 x π x 12 or 12 + π x 12 (implied by rounding to18.8 or (ii) rounds to 56.5 or 56.6 2 18.9 or 49.7 seen) M1 for using π x 62 (implied by 113 (. ) seen

This question in 0580/11 Oct/Nov 2004

17 For Examiner's Use NOT TO SCALE 90 m (a) The diagram shows the plan for a new soccer field. The length of the pitch is 90 metres. The ratio length : width is 5 : 3. Calculate the width of the pitch. Answer(a) m [2] (b) The centre circle has a circumference of 57.5 metres. Calculate the radius. Answer(b) m [2]

4 marks

Mark scheme: 17 (a) 54 2 M1 for 90 ÷ 5 x 3 (b) 9.15(….) 2 M1 for 57.5 ÷ (2π) or SC1 for 57.5 ÷ π. (implied by 18.3…)

This question in 0580/11 May/June 2006

17 The surface area of a sphere with radius r is A = 4πr2. (a) Calculate the surface area of a sphere with a radius of 5 centimetres. Answer(a) cm2 [1] (b) Make r the subject of the formula A = 4πr2. Answer(b) r = [2]

3 marks

Mark scheme: 17 (a) art 314 1 A A (b) oe 2 M1 for seen 4π 4π

This question in 0580/11 Oct/Nov 2006

9 Calculate the area of a circle of radius 3.75 cm. Answer cm2 [2]

2 marks

Mark scheme: 9 44.2 2 M1 for 3.752 × π or 44.15 to 44.19

This question in 0580/13 May/June 2010

8 A circle has a radius of 50 cm. (a) Calculate the area of the circle in cm2. Answer(a) cm2 [2] (b) Write your answer to part (a) in m2. Answer(b) m2 [1]

3 marks

Mark scheme: 8 (a) 7853 to 7855 2 M1 for π × 502 or 7850 or 7860 www (b) 0.7853 to 0.7855 or 0.785 or 0.786 1ft Their (a) ÷ 10 000 evaluated

This question in 0580/11 Oct/Nov 2011

12 NOT TO 6 cm SCALE The diagram shows a circular disc with radius 6 cm. In the centre of the disc there is a circular hole with radius 0.5 cm. Calculate the area of the shaded section. Answer cm2 [3]

3 marks

Mark scheme: 12 112 or 112.3 to 112.33 3 M2 for π × 62 − π × 0.52 or M1 for π × 62 or π × 0.52 seen

This question in 0580/11 Oct/Nov 2012

15 The table shows how 45 students each travel to college. For Examiner's Use Method of travel Walk Bus Cycle Frequency 20 18 7 This information can be displayed in a pie chart. (a) Show that the sector angle for students who walk is 160°. Answer(a) [1] (b) Calculate the sector angle for students who travel by bus. Answer(b) [1] (c) Complete the pie chart and label the sectors. [2]

4 marks

Mark scheme: 0 15 (a) × 360 [ = 160] 1 45 (b) 144 1 (c) Pie chart with at least 2 correct 2 B1 for a sector of 158° to 162° or 142° to 146° sectors and at least 2 sectors or 54° to 58° correctly labelled. (a) 0  1 1

This question in 0580/11 Oct/Nov 2012

20 B C NOT TO O SCALE A A, B and C are points on the circumference of a circle centre O. AC is a straight line. (a) Explain why angle ABC is 90°. Answer(a) … [1] (b) The diameter of the circle is 3 cm. Calculate the area of this circle. Answer(b) … cm2 [2] _____________________________________________________________________________________

3 marks

Mark scheme: 20 (a) Angle or triangle [in a] semi-circle 1 (b) 7.068 to 7.07 2 M1 for π × 1.52 seen 2 3  

This question in 0580/12 May/June 2013

20 For Examiner′s B Use NOT TO SCALE 17 cm C 6 cm A In the diagram, AB is a diameter of the circle and C is a point on the circumference. AB = 17 cm and AC = 6 cm. (a) Calculate the area of the circle. Answer(a) … cm2 [2] (b) (i) Explain why angle ACB = 90°. Answer(b)(i) … [1] (ii) Calculate BC. Answer(b)(ii) BC = … cm [3] _____________________________________________________________________________________

6 marks

Mark scheme: 20 (a) 226.98 to 227.01 2 M1 for π × (17 ÷ 2) 2 (b) (i) Angle or triangle [in a] semi-circle 1 (ii) 15.9 or 15.90 to 15.91 3 2 2 M2 for 17 − 6 or or 253 2 2 2 M1 for 17 = BC + 6 or better.

This question in 0580/13 May/June 2013

7 Find the circumference of a circle of radius 2.5 cm. Answer … cm [2] _____________________________________________________________________________________

2 marks

Mark scheme: 7 15.7 or 15.70 to 15.71 2 M1 for 2 × π × 2.5 8

This question in 0580/12 Oct/Nov 2013

21 The diagram shows a circle inside a square. The circumference of the circle touches all four sides of the square. (a) Calculate the area of the circle when the side of the square is 15 cm. Answer(a) … cm2 [2] (b) Draw all the lines of symmetry on the diagram. [2] __________________________________________________________________________________________ Question 22 is printed on the next page.

4 marks

Mark scheme: 21 (a) 177 or 176.7 to 176.74 2 M1 for π × 7.52 (b) 4 correct lines of symmetry drawn 2 B1 for 2 correct and no extra lines 27

This question in 0580/11 May/June 2014

13 The four sector angles in a pie chart are 2x°, 3x°, 4x° and 90°. Find the value of x. Answer x = … [2] __________________________________________________________________________________________

2 marks

Mark scheme: 13 30 2 M1 for 2 x + 3x + 4 x + 90 = 360 oe

This question in 0580/13 Oct/Nov 2014

22 (a) O O is the centre of the circle. Measure the diameter of this circle. Give your answer in millimetres. Answer(a) … mm [1] (b) A circular dinner plate has radius 12.7 cm. Work out the area of the plate. Answer(b) … cm2 [2]

3 marks

Mark scheme: 22 (a) 44 to 48 1 (b) 507 or 506.7 to 506.8 2 M1 for π×12.72

This question in 0580/12 Oct/Nov 2015

22 A circle has a radius of 6.4 cm. (a) Work out the circumference of the circle. … cm [2] (b) The circle forms the top of a cylinder of height 12 cm. Work out the volume of the cylinder. … cm3 [2] Question 23 is printed on the next page.

4 marks

Mark scheme: 22 (a) 40.2 or 40.21 to 40.22 2 M1 for 2 × π × 6.4 oe (b) 1540 or 1544 2 M1 for π × 6.42 × 12 or 1544.1 to 1544.4

This question in 0580/13 Oct/Nov 2016

18 The diagram shows a pentagon ABCDE. A E B D C (a) Using a straight edge and compasses only, construct the bisector of angle BCD. [2] (b) Draw the locus of the points inside the pentagon that are 3 cm from E. [1] (c) Shade the region inside the pentagon that is • less than 3 cm from E and • nearer to DC than to BC. [1]

4 marks

Mark scheme: 18(a) Correct ruled bisector with two pairs of 2 B1 for correct ruled bisector with arcs no/wrong arcs 18(b) Correct arc centre E radius 3 cm inside 1 pentagon 18(c) Correct region shaded 1 Dependent on at least B1 in part (a) and 1 mark in part (b) and a closed region

This question in 0580/12 Oct/Nov 2018

13 Calculate the area of a circle with radius 12 cm. … cm2 [2]

2 marks

Mark scheme: 13 452 or 452.3 to 452.4... 2 M1 for 122 × π

This question in 0580/11 Oct/Nov 2019

10 Calculate the circumference of a circle with radius 4.5 cm. … cm [2]

2 marks

Mark scheme: 10 28.3 or 28.27 to 28.28 2 M1 for 2 × π × 4.5 oe

This question in 0580/13 Oct/Nov 2019

23 The scale drawing shows a triangle ABC. The scale is 1 centimetre represents 8 kilometres. B A C Scale : 1 cm to 8 km (a) Using a straight edge and compasses only, construct the bisector of angle BAC. Show all your construction arcs. [2] (b) Draw the locus of points inside triangle ABC that are 56 km from C. [2] (c) Shade the region inside triangle ABC that is • more than 56 km from C and • nearer to AC than to AB. [1]

5 marks

Mark scheme: 23(a) correct line with two correct pairs 2 B1 for correct line with no or incorrect arcs of correct arcs 23(b) arc centre C radius 7 cm and from 2 B1 for any arc centre C from AC to BC or arc AC to BC radius 7 cm from C not reaching AC and/or BC 23(c) correct region shaded 1 Dep on at least B1 B1

This question in 0580/13 Oct/Nov 2019

2 A B (a) Measure the length of the line AB in millimetres. … mm [1] (b) AB is the diameter of a circle. Draw this circle. [2]

3 marks

Mark scheme: 2(a) 94 1 2(b) Correct circle drawn 2 B1 for centre of AB marked

This question in 0580/12 Feb/March 2020

19 A circle has a circumference of 56 mm. Work out the radius of this circle. … mm [2]

2 marks

Mark scheme: 19 8.91 or 8.911 to 8.913 2 56 M1 for 2π

This question in 0580/11 Oct/Nov 2020

13 The radius of a circle is 42 cm. Work out the circumference of the circle. Give your answer in terms of r. … cm [2]

2 marks

Mark scheme: 13 84π final answer 2 M1 for 42 × 2 × π soi

This question in 0580/11 May/June 2021

25 B NOT TO SCALE 72° O 9 cm A The diagram shows a sector of a circle, centre O, radius 9 cm. The sector angle is 72°. (a) Calculate the length of the arc AB. … cm [2] (b) Calculate the area of the sector AOB. … cm2 [2]

4 marks

Mark scheme: 25(a) 11.3 or 11.305 to 11.312 2 72 M1 for × 2 × 9 × π 360 25(b) 50.9 or 50.89 to 50.91 2 72 2 M1 for × 9 × π 360

This question in 0580/12 May/June 2021

2 NOT TO SCALE O B A Points A and B lie on a circle, centre O. (a) Write down the mathematical name for line AB. … [1] (b) The circle has a diameter of 16.8 cm. Write down the radius of the circle. … cm [1]

2 marks

Mark scheme: 2(a) Chord 1 2(b) 8.4 oe 1

This question in 0580/13 May/June 2021

13 A circular disc has circumference 250 cm. Calculate the radius of the disc. … cm [2]

2 marks

Mark scheme: 13 39.8 or 39.78 to 39.79 2 M1 for 250 ÷ 2π

This question in 0580/11 Oct/Nov 2021

21 45° NOT TO 4.8 cm SCALE The diagram shows a sector of a circle with radius 4.8 cm and sector angle 45°. Calculate the area of the sector. … cm2 [2]

2 marks

Mark scheme: 21 9.05 or 9.047 to 9.049 2 2 45 M1 for π × 4.8 × oe 360

This question in 0580/11 Oct/Nov 2021

19 Calculate the radius of a circle with circumference 26 cm. … cm [2]

2 marks

Mark scheme: 19 4.14 or 4.137 to 4.138... 2 26 M1 for 2π

This question in 0580/13 Oct/Nov 2021

3 (a) The diagram shows a circle. On the diagram, draw a chord. [1] (b) Another circle has a diameter of 28 cm. Find the radius of this circle. … cm [1]

2 marks

Mark scheme: 3(a) Chord drawn 1 3(b) 14 1

This question in 0580/11 May/June 2022

9 Thibault records the number of cars of each colour in a car park. Colour Black White Silver Red Number of cars 8 5 4 3 (a) He draws a pie chart to show this information. Calculate the sector angle for the red cars. … [2] (b) Two more white cars enter the car park and no cars leave the car park. When these two white cars are included in the results, will the sector angle for the red cars change? Without doing any further calculations, give a reason for your decision. … because … [1]

3 marks

Mark scheme: 9(a) 54 2 360 3 M1 for [ 3] or [ 360] oe 8  5  4  3 8  5  4  3 9(b) Yes [because] the proportion of 1 red cars will be smaller oe

This question in 0580/12 May/June 2022

16 The circumference of a circular pond is 130 cm. Calculate the radius of the pond. … cm [2]

2 marks

Mark scheme: 16 20.7 or 20.68 to 20.69[0…] 2 M1 for 2  π  r = 130 or better

This question in 0580/13 Oct/Nov 2022

26 A circle has an area of 25rcm 2. (a) Work out the circumference of the circle. Give your answer in terms of r. … cm [3] (b) Two of the circles are used as the ends of a cylinder, with height h cm. The total surface area of the cylinder is 170rcm 2. Work out the value of h. h = … [3]

6 marks

Mark scheme: 26(a) 10π 3 M1 for 25 π = πr 2 or B1 for r = 5 M1dep for 2π  theirr 26(b) 12 3 M2 for 25π +2 their10π =h 170π or better or M1 for 2  πr 2 + 2πrh = 170π or M1 for 170π −2 25π or better

This question in 0580/12 Feb/March 2023

17 A B NOT TO O SCALE C A, B and C are points on a circle, centre O. (a) Draw a tangent to the circle at point A. [1] (b) The circumference of the circle is 22.3 cm. Calculate the radius of the circle. … cm [2] (c) Give a geometrical reason why angle BCA is 90°. … [1]

4 marks

Mark scheme: 17(a) Ruled tangent drawn at A 1 17(b) 3.55 or 3.548 to 3.549… 2 M1 for [r =] 22.3 ÷ 2π 17(c) Angle in a semicircle is 90º oe 1

This question in 0580/12 Oct/Nov 2023

20 72° r cm NOT TO SCALE The diagram shows a sector of a circle with radius r cm and sector angle 72°. The arc length is 9.35 cm. Calculate the value of r. r = … [2]

2 marks

Mark scheme: 20 7.44 or 7.439 to 7.440… 2 72 M1 for 2 × π × r × = 9.35 oe or better 360

This question in 0580/13 Oct/Nov 2023

19 NOT TO SCALE 7 cm R cm The diagram shows a small circle with radius 7 cm and a large circle with radius R cm. The area of 16 small circles is the same as the area of one large circle. Calculate the value of R. R = … [3]

3 marks

Mark scheme: 19 28 3 M2 for 16  7 2 = R 2 or better or M1 for 7 2

This question in 0580/12 Feb/March 2024

17 The diagram shows a circle, centre O. P lies on the circle. P O (a) Write down the mathematical name of the line OP. … [1] (b) Draw a tangent to the circle at P. [1]

2 marks

Mark scheme: 17(a) radius 1 17(b) Ruled line drawn through P 1 perpendicular to OP

This question in 0580/11 May/June 2024

16 A circle has an area of 36r cm 2 . (a) Find the circumference of the circle. Give your answer in terms of r. … cm [3] (b) The circle forms the base of a cylinder with height h cm. The volume of the cylinder is 540r cm 3 . Work out the value of h. h = … [2]

5 marks

Mark scheme: 16(a) 12π cao 3 B2 for r = 6 or M1 for πr2 = 36π oe 16(b) 15 cao 2 M1 for [h =] 540π ÷ 36π or better

This question in 0580/12 May/June 2024

25 The diagram shows a shape made from two different semicircles, with the same centre. NOT TO SCALE 4 cm 7 cm The radius of the large semicircle is 7 cm. The radius of the small semicircle is 4 cm. Work out the perimeter of the shape. Give your answer in terms of r. … cm [3]

3 marks

Mark scheme: 25 11π + 6 final answer 3 B2 for 11π + ,k k 0 as final answer or for correct answer seen and spoilt or M2 for 7π + 4π + 2  ( 7 − 4 ) oe or M1 for 7π or 4π or [2×](7 – 4) oe

This question in 0580/12 May/June 2025

19 60° NOT TO SCALE 3 cm The diagram shows a sector of a circle with radius 3 cm and sector angle 60°. Calculate the area of the sector. Give your answer in terms of r in its simplest form. … cm2 [2]

2 marks

Mark scheme: 19 3 1 2 60 2 π , 1 π , or 1.5π M1 for π 3 oe 2 2 360

This question in 0580/13 May/June 2025

2 O The diagram shows a circle, centre O. On the diagram (a) draw a diameter [1] (b) mark a point on the circumference and label it C. [1]

2 marks

Mark scheme: 2(a) Diameter drawn 1 2(b) Point marked on circumference 1

This question in 0580/11 Oct/Nov 2025

18 The diameter of a circle is 16 cm. Find the area of the circle. Leave your answer in terms of r. … cm2 [2]

2 marks

Mark scheme: 18 64π 2 2  16  M1 for    2 

This question in 0580/13 Oct/Nov 2025