2022 May/June · Paper 11 · Q4 (a, b)3 marksNets & 3D shapes · Surface area & volume
2022 May/June · Paper 11 · Q53 marksNets & 3D shapes
2022 May/June · Paper 12 · Q72 marksArea & perimeter
2022 May/June · Paper 12 · Q232 marksArcs & sectors
2022 May/June · Paper 21 · Q4 (a, b(i), b(ii), c)8 marksArcs & sectors · Surface area & volume
2022 May/June · Paper 22 · Q4 (a(i), b, c(i), c(ii))11 marksArea & perimeter · Surface area & volume
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school. “Paper 42” means Paper 4, variant 2.
Practice questions
Original questions written by HomiLabs in the style of the exam, with answers and mark schemes. They are not
Cambridge past paper questions and have not been reviewed by Cambridge. Try each one before opening the answer.
Q1Surface area & volume · Core · Paper 2 style · calculator · 5 marks
A closed cylindrical tin has radius 4 cm and height 10 cm.
(a)Find the volume of the tin, correct to 3 significant figures.[2]
(b)Find the total surface area of the closed tin, correct to 3 significant figures.[3]
Q2Surface area & volume · Core · Paper 2 style · calculator · 4 marks
(a)A water tank on a roof in Hyderabad is a cuboid measuring 1.2 m by 0.8 m by 0.6 m. Calculate the volume of the tank in m³.[1]
(b)Write the volume in litres (1 m³ = 1000 litres).[1]
(c)The empty tank is filled from a pipe at 12 litres per minute. How long does it take to fill? Give your answer in minutes.[2]
Show answer
(a) 0.576 m³
1.2 × 0.8 × 0.6 B1
(b) 576 litres
× 1000 B1
(c) 48 minutes
576 ÷ 12 M1 A1
Q3Surface area & volume · Core · Paper 1 style · no calculator · 6 marks
(a)A tent is a triangular prism of length 3 m. Its cross-section is an isosceles triangle with base 2.4 m and perpendicular height 1.6 m. Find the volume of air inside the tent.[2]
(b)Show that each sloping side of the triangle is 2 m long.[1]
(c)The tent has a groundsheet, so all five faces are made of fabric. Find the total area of fabric.[3]
Show answer
(a) 5.76 m³
Area of triangle ½ × 2.4 × 1.6 = 1.92 M1; × 3 A1
(b) √(1.2² + 1.6²) = √4 = 2 m
Half base 1.2, Pythagoras √(1.44 + 2.56) B1
(c) 23.04 m²
Triangles 2 × 1.92 = 3.84; base 2.4 × 3 = 7.2; sloping faces 2 × 2 × 3 = 12 M2 (M1 for two of these); total A1
Q4Surface area & volume · Core · Paper 2 style · calculator · 4 marks
Give your answers correct to 3 significant figures.
(a)A football is a sphere of radius 11 cm. Calculate its volume. [Volume of sphere = 4/3 πr³][2]
(b)Calculate its surface area. [Surface area of sphere = 4πr²][2]
Show answer
(a) 5580 cm³
4/3 × π × 11³ M1 A1 (5575.28…)
(b) 1520 cm²
4 × π × 11² M1 A1 (1520.53…)
Q5Surface area & volume · Core · Paper 1 style · no calculator · 4 marks
A cylindrical tin of ghee has radius 7 cm and volume 245π cm³.
(a)Find the height of the tin.[2]
(b)Find the curved surface area of the tin in terms of π.[2]
Show answer
(a) 5 cm
π × 7² × h = 245π M1; h = 5 A1
(b) 70π cm²
2π × 7 × 5 M1 A1
Q6Surface area & volume · Standard · Paper 1 style · no calculator · 6 marks
A solid is made from a hemisphere of radius 6 cm and a cone of radius 6 cm and height 8 cm, joined at their circular faces.
(a)Find the slant height of the cone.[1]
(b)Find the total volume of the solid. Give your answer in terms of π.[3]
(c)Find the total surface area of the solid. Give your answer in terms of π.[2]
Q7Surface area & volume · Standard · Paper 2 style · calculator · 6 marks
An ice-cream cone is a hollow cone with base radius 3.5 cm and slant height 12 cm. Give your answers correct to 3 significant figures.
(a)Calculate the vertical height of the cone.[2]
(b)Calculate the volume of the cone. [Volume of cone = ⅓πr²h][2]
(c)Calculate the curved surface area of the cone. [Curved surface area = πrl][2]
Show answer
(a) 11.5 cm
h² = 12² − 3.5² M1; h = √131.75 = 11.478… A1
(b) 147 cm³
⅓ × π × 3.5² × 11.478… M1 A1
(c) 132 cm²
π × 3.5 × 12 M1 A1
Q8Surface area & volume · Standard · Paper 1 style · no calculator · 3 marks
A cylindrical jar has internal radius 5 cm. It contains water to a depth of 8 cm. A solid metal ball of radius 3 cm is placed in the jar and is completely covered by the water. [Volume of sphere = 4/3 πr³]
(a)Find the volume of the ball. Give your answer in terms of π.[1]
(b)Find the rise in the water level.[2]
Show answer
(a) 36π cm³
4/3 × π × 27 B1
(b) 36/25 cm (1.44 cm)
π × 5² × h = 36π M1; h = 36/25 = 1.44 A1
Q9Surface area & volume · Standard · Paper 2 style · calculator · 5 marks
A copper pipe is a hollow cylinder 2 m long. Its external radius is 1.5 cm and its internal radius is 1.2 cm. Give your answers correct to 3 significant figures.
(a)Calculate the volume of copper in the pipe, in cm³.[3]
(b)Copper has a density of 8.9 g/cm³. Calculate the mass of the pipe in kilograms.[2]
Q10Surface area & volume · Standard · Paper 1 style · no calculator · 6 marks
A monument is a square-based pyramid. The base has side 30 m and the vertical height is 20 m. The top is directly above the centre of the base.
(a)Find the volume of the monument. [Volume of pyramid = ⅓ × base area × height][2]
(b)Show that the slant height of each triangular face is 25 m.[2]
(c)The four triangular faces are to be cleaned. Find their total area.[2]
Show answer
(a) 6000 m³
⅓ × 900 × 20 M1 A1
(b) √(20² + 15²) = √625 = 25
Half the base 15 M1; Pythagoras A1
(c) 1500 m²
4 × ½ × 30 × 25 M1 A1
Q11Surface area & volume · Standard · Paper 2 style · calculator · 5 marks
A swimming pool in Lahore is 25 m long and 10 m wide. The depth increases steadily from 1 m at the shallow end to 2.5 m at the deep end, so its side face is a trapezium.
(a)Calculate the volume of water in the pool when it is full.[3]
(b)Water is pumped in at 0.6 m³ per minute. How long does it take to fill the empty pool? Give your answer in hours and minutes, to the nearest minute.[2]
Show answer
(a) 437.5 m³
Cross-section is a trapezium ½(1 + 2.5) × 25 = 43.75 M1 A1; × 10 A1
(b) 12 hours 9 minutes
437.5 ÷ 0.6 = 729.17 min M1; 12 h 9 min A1
Q12Surface area & volume · Challenge · Paper 2 style · calculator · 6 marks
A solid metal cube of side 10 cm is melted down and made into solid balls, each of radius 1.5 cm.
(a)Find the volume of one ball, in terms of π.[1]
(b)Find the greatest number of balls that can be made, and the volume of metal left over, correct to 3 significant figures.[3]
(c)All 70 balls are dropped into a tall cylinder of radius 6 cm partly filled with water, and they are all covered by water. Find how much the water level rises.[2]
Q13Surface area & volume · Challenge · Paper 2 style · calculator · 5 marks
A bucket is a frustum: it is made by cutting a small cone off the top of a large cone, parallel to the base. The bucket has base radius 9 cm at the bottom, radius 15 cm at the top and depth 24 cm. The large cone has radius 15 cm and the small cone removed has radius 9 cm. [Volume of cone = ⅓πr²h]
(a)Show that the large cone has height 60 cm.[2]
(b)Calculate the capacity of the bucket in litres, correct to 3 significant figures.[3]
Show answer
(a) (h − 24)/h = 9/15 gives h = 60 cm
Similar cones: (h − 24)/h = 9/15 M1; 15h − 360 = 9h, h = 60 A1
Q15Surface area & volume · Challenge · Paper 2 style · calculator · 7 marks
A full cylindrical can of soup has internal radius 6 cm and height 15 cm. The soup is poured into identical hemispherical bowls of internal radius 5 cm.
(a)Calculate the volume of soup in the can, correct to 4 significant figures.[2]
(b)Calculate the volume of one bowl, correct to 4 significant figures. [Volume of sphere = 4/3 πr³][2]
(c)How many bowls can be filled completely? Find the volume of soup left over, correct to 3 significant figures.[3]
Q16Surface area & volume · Challenge · Paper 1 style · no calculator · 6 marks
A solid sphere has radius 6 cm. A solid cylinder has radius 4 cm, height h cm and the same volume as the sphere. [Volume of sphere = 4/3 πr³, surface area of sphere = 4πr²]
(a)Find the volume of the sphere in terms of π.[1]
(b)Find h.[2]
(c)Find, in terms of π, how much greater the total surface area of the cylinder is than the surface area of the sphere.[3]
Q21Arcs & sectors · Standard · Paper 2 style · calculator · 4 marks
A pendulum has a string of length 80 cm. In one swing, the bob moves along an arc of length 35 cm.
(a)Calculate the angle through which the string turns, correct to 1 decimal place.[2]
(b)Calculate the area of the sector swept out by the string, correct to 3 significant figures.[2]
Show answer
(a) 25.1°
θ/360 × 2π × 80 = 35 M1; θ = 25.07… A1
(b) 1400 cm²
Area = θ/360 × π × 80² (or ½ × arc × radius) M1; 1400 A1
Q22Arcs & sectors · Challenge · Paper 2 style · calculator · 7 marks
(a)A circle has radius 10 cm. Chord AB subtends an angle of 70° at the centre. Find the area of the minor segment, correct to 3 significant figures.[3]
(b)Another sector has arc length 15 cm and area 60 cm². Find its radius and its angle, correct to 1 decimal place.[4]
Q29Nets & 3D shapes · Core · Paper 1 style · no calculator · 3 marks
(a)Write down the number of faces, edges and vertices of a cube.[1]
(b)Write down the number of faces, edges and vertices of a square-based pyramid.[2]
Show answer
(a) 6 faces, 12 edges, 8 vertices
B1.
(b) 5 faces, 8 edges, 5 vertices
All three B2; two correct B1.
Q30Nets & 3D shapes · Standard · Paper 1 style · no calculator · 5 marks
(a)A net of a cuboid 5 cm by 4 cm by 3 cm is cut from card. How many rectangles does the net have and what is its total area?[3]
(b)The net of a closed cylinder of radius 3 cm and height 10 cm is two circles and a rectangle. Write down the length and width of the rectangle. Give the length in terms of π.[2]
Show answer
(a) 6 rectangles; 94 cm²
6 B1; 2(5×4 + 5×3 + 4×3) M1; 94 A1.
(b) 6π cm by 10 cm
Length = circumference 6π B1; width 10 B1.
Q31Nets & 3D shapes · Challenge · Paper 2 style · calculator · 5 marks
The net of the curved surface of a paper cone is a sector of radius 15 cm and angle 216°. The two straight edges are joined with no overlap.
(a)Find the radius of the base of the cone.[2]
(b)Find the height of the cone.[1]
(c)Find the volume of the cone, correct to 3 significant figures.[2]
