Surface area and volume — Paper 2 · IGCSE Mathematics 0580

13 (a) Write down the number of lines of symmetry for the diagram below. Answer(a) [1] (b) Write down the order of rotational symmetry for the diagram below. Answer(b) [1] (c) The diagram shows a cuboid which has no square faces. Draw one of the planes of symmetry of the cuboid on the diagram. [1]

3 marks

Mark scheme: 13 (a) 0 1 (b) 2 1 (c) plane across centre of shape 1 Three possibilities

This question in 0580/23 May/June 2010

8 Calculate the radius of a sphere with volume 1260 cm3 . ForFor 4 Examiner'sExaminer's [The volume, V, of a sphere with radius r is V = πr3.] UseUse 3 Answer cm [3]

3 marks

Mark scheme: 3 8 6.70 3 M1 for ( r3 = ) 1260 × oe seen 4 π M1 for 3 of their r3 seen or implied

This question in 0580/21 May/June 2011

17 NOT TO SCALE 20 cm 10 cm 9 cm d cm The diagrams show two mathematically similar containers. The larger container has a base with diameter 9 cm and a height 20 cm. The smaller container has a base with diameter d cm and a height 10 cm. (a) Find the value of d. Answer(a) d = [1] (b) The larger container has a capacity of 1600 ml. Calculate the capacity of the smaller container. Answer(b) ml [2]

3 marks

Mark scheme: 17 (a) 4.5(0) 1 (b) 200 2 M1 0.53 or 23 seen

This question in 0580/23 May/June 2011

21 For P Examiner's Use NOT TO SCALE 4 cm C D 6 cm M A 6 cm B The diagram shows a pyramid with a square base ABCD of side 6 cm. The height of the pyramid, PM, is 4 cm, where M is the centre of the base. Calculate the total surface area of the pyramid. Answer cm2 [5] Question 22 is printed on the next page.

5 marks

Mark scheme: 21 96 www 5 M1 32 + 42 A1 5 M1 ½ × 6 × “5” (= 15) M1 4 × their triangle area + 62 IGCSE – May/June 2011 0580 23 22 (a) 159 3 M1 evidence of using area under graph M1 stating area correctly

This question in 0580/23 May/June 2011

14 For Examiner's Use NOT TO SCALE r 2r The sphere of radius r fits exactly inside the cylinder of radius r and height 2r. Calculate the percentage of the cylinder occupied by the sphere. 4 [The volume, V, of a sphere with radius r is V = πr3.] 3 Answer % [3]

3 marks

Mark scheme: 2 3 π 2 14 66 or 66.7 www 3 M2 for (× 100) or M1 for πr ( 2 r ) 3 πr 2 ( 2 r )

This question in 0580/23 Oct/Nov 2011

16 For Examiner's NOT TO Use SCALE 4 cm 15 cm The diagram shows a solid prism of length 15 cm. The cross-section of the prism is a semi-circle of radius 4 cm. Calculate the total surface area of the prism. Answer cm2 [4]

4 marks

Mark scheme: 16 359 www 4 M1 π × 42 or 12 π × 4 2 M1 for 0.5 × π × 8 × 15 oe M1 for 8 × 15 + their 2 ends + their curved surface area

This question in 0580/23 May/June 2012

15 A sphere has a volume of 80 cm3. For Examiner′s Use Calculate the radius of the sphere. 4 [The volume, V, of a sphere with radius r is V = 3 πr 3 .] Answer … cm [3] _____________________________________________________________________________________

3 marks

Mark scheme: 15 2.67 or 2.672 to 2.67301 3 M2 for 3 (80 ÷ 43 π ) oe or M1 for 80 ÷ ( 43π ) oe

This question in 0580/21 May/June 2013

26 For Examiner′s Use NOT TO C SCALE D 6 cm 4 cm 15 cm A 9 cm B The diagram shows a solid prism of length 15 cm. The cross section of the prism is the trapezium ABCD. Angle DAB = angle CDA = 90°. AB = 9 cm, DC = 6 cm and AD = 4 cm. Calculate the total surface area of the prism. Answer … cm2 [5]

5 marks

Mark scheme: 26 420 5 M1 for [CB =] 4 2 + (9 − 6) 2 M1 for their CB from Pythagoras × 15 M1 for [2 ×] 12 ( 6 + 9) × 4 M1 for 4 × 15, 9 × 15, 6 × 15 with intention to add

This question in 0580/21 May/June 2013

18 NOT TO SCALE 10 cm 4 cm A solid cone has base radius 4 cm and height 10 cm. A mathematically similar cone is removed from the top as shown in the diagram. 1 The volume of the cone that is removed is of the volume of the original cone. 8 (a) Explain why the cone that is removed has radius 2 cm and height 5 cm. Answer(a) [2] (b) Calculate the volume of the remaining solid. 1 [The volume, V, of a cone with radius r and height h is V = πr 2h.] 3 Answer(b) … cm3 [4] __________________________________________________________________________________________ Question 19 is printed on the next page.

6 marks

Mark scheme: 10 4 1 1 = = 2 or 3 8 = 2 AND = 5 oe and18 (a) correct working 2 B2 for 3 8 2 2 2 oe or 1 1 1 = ( 3) or 3 8 or 8 = 23 or B1 for 3 8 8 2 IGCSE – May/June 2014 0580 21 7 1 2 (b) 147 or 146.5 to 146.6… 4 M3 for × × π × 4 × 10 8 3 or 1 2 M1 for × π × 4 × 10 3 and 1 2 M1 for × π × 2 × 5 3 and M1 for subtracting their volumes 1 i

This question in 0580/21 May/June 2014

17 The diagram shows a child’s toy. 5 cm NOT TO SCALE 8 cm The shape of the toy is a cylinder of radius 5 cm and height 8 cm on top of a hemisphere of radius 5 cm. Calculate the volume of the toy. 4 [The volume, V, of a sphere with radius r is V = πr 3.] 3 Answer … cm3 [5] __________________________________________________________________________________________

5 marks

Mark scheme: 1  4 3  217 890 or 890.1 to 890.2… 5 M4 for ×  × π × 5  + π × 5 × 8 2  3  1  4 3  2 or M3 for ×  × π × 5  and π × 5 × 8 2  3  1  4 3  2 or M2 for ×  × π × 5  or π × 5 × 8 2  3  4 3 or M1 for × π × 5 3

This question in 0580/21 Oct/Nov 2014

21 NOT TO SCALE 9 cm 4 cm The diagram shows a toy. The shape of the toy is a cone, with radius 4 cm and height 9 cm, on top of a hemisphere with radius 4 cm. Calculate the volume of the toy. Give your answer correct to the nearest cubic centimetre. [The volume, V, of a cone with radius r and height h is V = 13 πr 2h.] [The volume, V, of a sphere with radius r is V = 43 πr 3.] Answer … cm3 [4] __________________________________________________________________________________________

4 marks

Mark scheme: 21 285 cao 4 M1 for 13 × π × 4 2 × 9 , 48π 1 4 3 128 π M1 for × × π × 4 , 2 3 3 272 π A1 for 284.8 to 284.9, 3 If A0 then B1 for their final answer rounded correctly to nearest whole number from their more accurate answer dependent on at least M1  22 17 

This question in 0580/21 May/June 2015

18 NOT TO SCALE 8 m 5 m 5 m 12 m The diagram shows the front face of a barn. The width of the barn is 12 m. The height of the barn is 8 m. The sides of the barn are both of height 5 m. (a) Work out the area of the front face of the barn. Answer(a) … m2 [3] (b) The length of the barn is 15 m. NOT TO Work out the volume of the barn. SCALE 15 m Answer(b) … m3 [1] __________________________________________________________________________________________

4 marks

Mark scheme: 1 18 (a) 78 3 M2 for 5 × 12 + × 12 × (8 – 5) or 2 1 × 6 × (5 + 8) × 2 oe 2 1 or M1 for 5 × 12, × 12 × (8 – 5) , 2 1 × 6 × (5 + 8) or 12 × 8 – (…) 2 (b) 1170 1FT 15 × their (a)

This question in 0580/22 May/June 2015

15 A solid consists of a metal cube with a hemisphere cut out of it. 5 cm NOT TO SCALE 7 cm The length of a side of the cube is 7 cm. The diameter of the hemisphere is 5 cm. Calculate the volume of this solid. 4 rr3 .] [The volume, V, of a sphere with radius r is V = 3 … cm3 [3]

3 marks

Mark scheme: 3 1 4  5  15 310 or 310.2 to 310.3 3 M2 for 7 − × × π ×  2 3  2  3 1 4  5  or M1 for × × π ×  2 3  2  3 3 4  5  or SC1 for 7 − × π ×  soi 3  2  2

This question in 0580/23 May/June 2016

21 (a) B 4.8 cm NOT TO E 6 cm SCALE 9 cm k cm C D A Triangles CBA and CED are similar. AB is parallel to DE. AB = 9 cm, BE = 4.8 cm, EC = 6 cm and ED = k cm. Work out the value of k. k = … [2] (b) h NOT TO 20 cm SCALE Vase A Vase B The diagram shows two mathematically similar vases. Vase A has height 20 cm and volume 1500 cm3. Vase B has volume 2592 cm3. Calculate h, the height of vase B. h = … cm [3]

5 marks

Mark scheme: 9 6 + 4.8 21 (a) 5 2 M1 for = oe k 6 2592 (b) 24 3 M2 for 3 × 20 oe 1500 2592 1500 or M1 for 3 or 3 1500 2592

This question in 0580/23 May/June 2016

12 NOT TO SCALE 5 cm The diagram shows a hemisphere with diameter 5 cm. Calculate the volume of this hemisphere. [The volume, V, of a sphere with radius r is V = 43 r r 3.] … cm3 [2]

2 marks

Mark scheme:  1  4  5  ×12 32.7 or 32.72 to 32.73 2 M1 for × π×     2  3  2 

This question in 0580/23 Oct/Nov 2016

5 Calculate the volume of a hemisphere with radius 3.2 cm. 4 [The volume, V, of a sphere with radius r is V = rr3 .] 3 … cm3 [2]

2 marks

Mark scheme: 5 68.6 or 68.62 to 68.64 2 1 4 3 M1 for × π × 3.2 2 3 If zero scored, SC1 for final answer 137 or 137.2 to 137.3

This question in 0580/23 May/June 2017

14 NOT TO SCALE 1 cm 7 cm The diagram shows a solid cuboid with base area 7 cm2. The volume of this cuboid is 21 cm3. Work out the total surface area. … cm2 [3]

3 marks

Mark scheme: 14 62 3 M1 for [height = ] 21 ÷ 7 M1 for 2(1 × their3 + their3 × 7 + 1 × 7) oe

This question in 0580/22 May/June 2018

15 Find the volume of a cylinder of radius 5 cm and height 8 cm. Give the units of your answer. … … [3]

3 marks

Mark scheme: 15 628 or 628.3 to 628.4 3 B2 for 628 or 628.3 to 628.4 or M1 for 52 × 8 × π cm3 B1 for cm3

This question in 0580/22 May/June 2018

25 (a) A NOT TO SCALE P Q B C In the diagram, PQ is parallel to BC. APB and AQC are straight lines. PQ = 8 cm, BC = 10 cm and AB = 9 cm. Calculate PB. PB = … cm [2] (b) 13 cm NOT TO SCALE The diagram shows two glasses which are mathematically similar. The larger glass has a capacity of 0.5 litres and the smaller glass has a capacity of 0.25 litres. The height of the larger glass is 13 cm. Calculate the height of the smaller glass. … cm [3]

5 marks

Mark scheme: 25(a) 1.8 2 10 9 M1 for = oe 8 AP 25(b) 10.3 or 10.31 to 10.32 3 0.25 M2 for 13 × 3 oe 0.5 0.5 0.25 0.5  13  3 or M1 for 3 oe or 3 oe or =   oe 0.25 0.5 0.25  h 

This question in 0580/23 May/June 2018

10 A water tank in the shape of a cuboid has length 1.5 metres and width 1 metre. The water in the tank is 60 centimetres deep. Calculate the number of litres of water in the tank. … litres [3]

3 marks

Mark scheme: 10 900 3 150 × 100 × 60 M2 for oe 1000 or M1 for 150 × 100 × 60 or 1.5[× 1] × 0.6 or B1 for figs 9

This question in 0580/21 Oct/Nov 2018

18 A pipe is full of water. The cross-section of the pipe is a circle, radius 2.6 cm. Water flows through the pipe into a tank at a speed of 12 centimetres per second. Calculate the number of litres that flow into the tank in one hour. … litres [3]

3 marks

Mark scheme: 18 917 or 918 or 917.4 to 917.6 3 M2 for π × 2.6 2 × 12 × 60 × 60 ÷ 1000 or M1 for π × 2.62 isw or 12 × 60 × 60 ÷ 1000 isw If 0 scored SC1 for figs 917 to 918

This question in 0580/22 Feb/March 2019

5 The volume of a cuboid is 180 cm3. The base is a square of side length 6 cm. Calculate the height of this cuboid. … cm [2]

2 marks

Mark scheme: 5 5 2 M1 for 180 ÷ 62 oe

This question in 0580/21 May/June 2019

9 A cuboid measures 5 cm by 7 cm by 9.5 cm. NOT TO SCALE 7 cm 5 cm 9.5 cm Work out the surface area of this cuboid. … cm2 [3]

3 marks

Mark scheme: 9 298 3 M2 for [2 ×] (5 × 7 + 5 × 9.5 + 7 × 9.5) oe or M1 for one correct area, 5 × 7 or 5 × 9.5 or 7 × 9.5

This question in 0580/23 Oct/Nov 2019

22 A container is made from a cylinder and a cone, each of radius 5 cm. The height of the cylinder is 12 cm and the height of the cone is 4.8 cm. 4.8 cm NOT TO 12 cm SCALE 5 cm The cylinder is filled completely with water. The container is turned upside down as shown below. NOT TO SCALE d Calculate the depth, d, of the water. 1 2 [The volume, V, of a cone with radius r and height h is V = r r h. ] 3 d = … cm [5] Question 23 is printed on the next page.

5 marks

Mark scheme: 22 15.2 5 M4 for  2 1 2  2 π × 5 × 12 − × π × 5 × 4.8 ÷ π × 5   ( )  3  2 1 2 or M3 for π × 5 × 12 − × π × 5 × 4.8 3 or M1 for π × 52 × 12 1 2 M1 for × π × 5 × 4.8 3

This question in 0580/23 Oct/Nov 2019

2 6.5 cm NOT TO SCALE 4 cm 8 cm The diagram shows a cuboid. Calculate the volume of the cuboid. … cm3 [1]

1 marks

Mark scheme: 2 208 1

This question in 0580/23 May/June 2020

20 A model of a statue has a height of 4 cm. The volume of the model is 12 cm3. The volume of the statue is 40 500 cm3. Calculate the height of the statue. … cm [3]

3 marks

Mark scheme: 20 60 3 40500 M2 for 4 × 3 oe 12  4 =3 12 oe or M1 for    l  40500 40500 12 or 3 oe or 3 oe 12 40500

This question in 0580/22 Oct/Nov 2020

17 Find the radius of a hemisphere of volume 80 cm 3 . 4 3 [The volume, V, of a sphere with radius r is V = rr . ] 3 … cm [3]

3 marks

Mark scheme: 17 3.37 or 3.367 to 3.368 3 3 120 M2 for isolating 3r , e.g. r   1 4 3 or M1 for   r  80 oe 2 3 If 0 scored SC1 for answer 2.67 or 2.672 to 2.673...

This question in 0580/21 May/June 2022

21 12.5 cm NOT TO SCALE 5.5 cm 9.2 cm A solid is made by cutting a small cone from a larger cone, as shown in the diagram. The height of the larger cone is 12.5 cm. The height of the solid is 5.5 cm. The diameter of the base of the larger cone is 9.2 cm. Work out the volume of the solid. 1 2 [The volume, V, of a cone with radius r and height h is V = rr h .] 3 … cm3 [4] Questions 22 and 23 are printed on the next page.

4 marks

Mark scheme: 21 228 or 228.3 to 228.4 4 2 1  9.2  M1 for π  12.5 oe   3  2  9.2 diameter M1 for = oe or better 12.5 12.5 − 5.5 1  their 5.152  2 M1 for π  (12.5 − 5.5 )   3  2  oe OR M2 for π  9.2  2 π 2   12.5 −  r  (12.5 − 5.5 ) oe   3  2  3 for any r < 4.6 If 0 scored SC1 for 913 or 913.3 to 913.5

This question in 0580/22 Oct/Nov 2022

6 Calculate the volume of a sphere with diameter 4.8 cm. 4 3 [The volume, V, of a sphere with radius r is V = rr .] 3 … cm3 [2]

2 marks

Mark scheme: 6 57.9 or 57.90 to 57.91... 2 3 4  4.8  M1 for    3  2 

This question in 0580/21 May/June 2023

3 NOT TO 5 cm SCALE 3 cm 8 cm Find the total surface area of the cuboid. … cm2 [3]

3 marks

Mark scheme: 3 158 3 M2 for 2 8  5 8 3 5 3  or M1 for 8  5 or 8  3 or 5  3

This question in 0580/23 May/June 2023

16 The volume of a cylinder is 1970 cm 3. The height of the cylinder is 12.8 cm . Calculate the radius of the cylinder. … cm [3]

3 marks

Mark scheme: 16 7.00 or 6.998 to 7.002 3 1970 M2 for [r2 ]= oe or better 12.8  or M1 for 1970   r 2  12.8 or better

This question in 0580/23 May/June 2023

11 A bronze sphere has radius 3.6 cm. The density of bronze is 8.05 g/cm3. Find the mass of the sphere. Give your answer in kilograms, correct to the nearest gram. 4 3 [The volume, V, of a sphere with radius r is V = rr .] 3 [Density = mass ' volume .] … kg [4]

4 marks

Mark scheme: 11 1.573 cao 4 B3 for answer figs 157[0] or 1573… OR 4 3 M2 for π 3.6  8.05 oe or better 3 4 3 or M1 for   3.6 oe 3 M1 for division by 1000 of their mass in g and correct rounding to 3 dp

This question in 0580/23 Oct/Nov 2023

16 A cylinder with height 12.5 cm has a curved surface area of 105r cm2 . Calculate the volume of the cylinder. … cm3 [4]

4 marks

Mark scheme: 16 693 or 692.7 to 692.8… 4 105 M2 for oe 2 12.5 or M1 for 2    r  12.5 = 105 or better M1 for   (their r)2  12.5

This question in 0580/23 Oct/Nov 2023

4 b cm NOT TO SCALE c cm d cm a cm Base The diagram shows the net of a cuboid with its base shaded. The length of the cuboid is 10 cm, its width is 4 cm and its height is 5 cm. Write down the values of each of a, b, c and d. a = … , b = … , c = … , d = … [4]

4 marks

Mark scheme: 4 a = 18 b = 10 c = 4 d = 9 4 B1 for each If 0 scored, SC1 for b or c = 4, 5 or 10

This question in 0580/22 Feb/March 2024

14 Two solid steel statues are mathematically similar. The smaller statue has height 12 cm and the larger statue has height 15 cm. The larger statue has a mass 2.5 kg. The density of steel is 8 g/cm3. Calculate the volume of the smaller statue. [Density = mass ÷ volume.] … cm3 [4]

4 marks

Mark scheme: 14 160 4 2500 12 3 M3 for V  = 3 oe 8 15 figs 25 12 3 or for answer figs 16 from × 8 15 3 or B2 for 1.28 [kg] OR M1 for 2500 ÷ 8 oe or 312.5 seen  12  3  15  3 M1 for   or   oe  15   12 

This question in 0580/21 Oct/Nov 2024

16 R cm NOT TO 6 cm SCALE 18 cm The diagram shows a sphere of radius 6 cm and a cylinder of height 18 cm and radius R cm. The volume of the sphere is equal to the volume of the cylinder. Calculate the curved surface area of the cylinder. Give your answer in terms of r. 4 3 [The volume, V, of a sphere with radius r is V = r r ] 3 … cm2 [4]

4 marks

Mark scheme: 16 144 cao 4 4 3  6 3 M2 for [ R2 =] oe 18  4 3 2 or M1 for   6 =  R  18 oe 3 M1 for 2  theirR 18 oe

This question in 0580/21 Oct/Nov 2024

19 4.3 cm NOT TO SCALE 11.9 cm A solid is made from a cylinder and a hemisphere, both of radius 4.3 cm. The cylinder has length 11.9 cm. (a) Calculate the volume of the solid. 4 3 [The volume, V, of a sphere with radius r is V = r r .] 3 … cm3 [3] (b) Calculate the total surface area of the solid. [The surface area, A, of a sphere with radius r is A = 4 rr 2 .] … cm2 [4]

7 marks

Mark scheme: 19(a) 858 or 857.7 to 857.9 3  1  4 3 M2 for   π  4.3 + π × 4.32 × 11.9 oe    2  3  1  4 3 or M1 for   π  4.3    2  3 or π × 4.32 × 11.9 19(b) 496 or 495.7 to 495.8… 4 M3 for 1 2 2 4  4.3 +  4.3 + 2  4.3 11.9 2 oe OR M1 for π × 4.32 × k where k is a whole number M1 for 2 × π × 4.3 × 11.9

This question in 0580/22 Oct/Nov 2024

13 r cm NOT TO SCALE 16 cm The diagram shows a cylinder with radius r cm and height 16 cm. A sphere has radius 3 cm. The volume of the cylinder is equal to the volume of the sphere. Find the value of r. r = … [4]

4 marks

Mark scheme: 13 3 4 B3 for 16r2 = 36 oe or better oe 2 2 4 3 or M2 for r 16 = 3 oe 3 4 3 or M1 for r 216 oe or 33 oe

This question in 0580/22 Feb/March 2025

13 4 cm 10 cm NOT TO h cm SCALE 6 cm Solid A Solid B The diagram shows solid A and solid B. Solid A is made from a hemisphere and a cone each with radius 6 cm. The cone has sloping edge 10 cm. Solid B is a cylinder with radius 4 cm and height h cm. The total surface area of solid A is equal to the total surface area of solid B. (a) Work out the value of h. h = … [5] (b) Work out the height of solid A. … cm [3]

8 marks

Mark scheme: 13(a) 12.5 oe 5 M4 for 8[]h = 100[] OR M3 for π 6 10 + 2 π 6 2 = 2 π 4 2 + 2 π 4 h oe OR M1 for π 6 10 + 2  π  6 2 oe M1 2 π 4 2 + 2 π 4 h oe OR SC2 for answer 16.5 13(b) 14 3 2 2 M2 for 10 − 6 or M1 for 62 + x2 = 102

This question in 0580/21 May/June 2025

16 NOT TO SCALE 6 cm 5 cm The diagram shows a solid made by joining a hemisphere to a cylinder. The radius of both the hemisphere and the cylinder is 6 cm. The height of the cylinder is 5 cm. Find the total surface area of the solid. Give your answer in terms of r. … cm2 [4]

4 marks

Mark scheme: 16 168 π 4 B3 for answer 168 OR M3 for 2 1 2 π  6 +  4π  6 + 2π 6 5 oe 2 OR To a maximum of 2 marks ignoring extra areas added or subtracted M1 for π × 62 1 2 M1 for 4 π 6 oe 2 M1 for 2 × π × 6 × 5

This question in 0580/22 May/June 2025

3 A cuboid has length 6 cm, width 5 cm and height 2.5 cm. Work out the volume of the cuboid. … cm3 [2]

2 marks

Mark scheme: 3 75 2 M1 for 6 × 5 × 2.5 oe

This question in 0580/22 Oct/Nov 2025

23 A NOT TO SCALE O 12 cm B 240° The diagram shows a major sector, AOB, of a circle. The sector angle is 240° and the radius is 12 cm. (a) Show that the length of the major arc AB is 16rcm . [1] (b) OA is joined to OB to form a cone. Work out the volume of the cone. `a bj r Give your answer in the form where a is an integer and b is a prime number. 3 … cm3 [6] Question 24 is printed on the next page.

7 marks

Mark scheme: 23(a) 240 M1 2 12 oe 360 23(b) 6 256 5  256 5 256 5  ( ) ( ) ( ) cao B5 for answer or 3 3 3 seen then spoilt 256 5  ( ) or an answer equivalent to 3 but not in required form OR B2 for radius of cone = 8 oe or M1 for 2πr = 16π B2 for [height of cone =] 80 oe or M1 for 122 = their 82 + h2 or better 1 2 M1dep for  their 8  their 80 oe 3 dep on attempt at Pythagoras to find height of cone

This question in 0580/22 Oct/Nov 2025