Edexcel IGCSE Maths
Topic questions Geometry and measures Volume and surface area
57 questions · 238 marks · about 5 hours
Also involves Bin packing
Boxes are put into a crate.
Each box is a cuboid, 16 cm by 4 cm by 6 cm The crate is a cuboid, 80 cm by 12 cm by 42 cm
Work out the greatest number of boxes that can be put into the crate.
Also involves Rearranging formulae
Here is a solid cylinder.
The radius of the cylinder is 8 cm The height of the cylinder is h cm
The volume of the cylinder is 3892 cm3
Work out the value of h Give your answer correct to one decimal place.
The diagram shows two water containers. One is a cuboid and one is a cylinder.
The cuboid measures 35 cm by 28 cm by 20 cm The surface of the water in the cuboid is 9 cm above the base of the cuboid.
The cylinder has a radius of 10 cm and a height of 33 cm The cylinder is completely full of water.
Izzy is going to pour all the water from the cylinder into the cuboid.
Show that the cuboid will not be completely full of water.
Here is a solid cylinder.
The radius of the cylinder is 8 cm The height of the cylinder is h cm
The volume of the cylinder is 3892 cm3
Work out the value of h Give your answer correct to one decimal place.
The diagram shows two water containers. One is a cuboid and one is a cylinder.
The cuboid measures 35 cm by 28 cm by 20 cm The surface of the water in the cuboid is 9 cm above the base of the cuboid.
The cylinder has a radius of 10 cm and a height of 33 cm The cylinder is completely full of water.
Izzy is going to pour all the water from the cylinder into the cuboid.
Show that the cuboid will not be completely full of water.
Also involves Forming and solving equations
The diagram shows a cuboid.
The cuboid measures 3x cm by 2x cm by y cm
The volume of the cuboid is 1014cm³ The total surface area of the cuboid is A cm²
You must show all the stages of your working.
The diagram shows a cylinder.
The radius of the cylinder is 70 cm The height of the cylinder is 18 cm
Work out the volume of the cylinder. Give your answer in litres correct to the nearest litre.
Also involves The four operations
The diagram shows a box and a crate with a lid.
The box is a cube with sides of length 5 cm. Tony has many boxes.
The crate is a cuboid with inside lengths of 27 cm, 35 cm and 40 cm.
Tony puts as many boxes as possible into the crate so that the lid will shut.
Work out the volume of space in the crate that is not filled up with boxes.
Also involves Converting between units
The diagram shows a solid wooden cylinder.
The cylinder has radius 8 cm and height h cm. The volume of the cylinder is 1208 cm3
Work out the value of h Give your answer correct to the nearest whole number.
The density of the wood is 1.25 g/cm³
Work out the mass of the cylinder. Give your answer in kilograms.
Also involves Rearranging formulae · Converting between units
The diagram shows a solid wooden cylinder.
The cylinder has radius 8 cm and height h cm. The volume of the cylinder is 1208 cm3
Work out the value of h Give your answer correct to the nearest whole number.
The density of the wood is 1.25 g/cm³
Work out the mass of the cylinder. Give your answer in kilograms.
Also involves Pythagoras' theorem
A solid shape is made by removing a hemisphere, shown shaded, from a cone as shown in the diagram.
The radius of the hemisphere is 2xcm The radius of the base of the cone is 5xcm The vertical height of the cone is 6xcm
The volume of the solid shape is 6948π cm3
Work out the total surface area of the solid hemisphere that has been removed from the cone. Give your answer correct to the nearest integer.
Here is a triangular prism ABCDEF
AD = 53 cm DF = 28 cm Angle FDE = 30°
Work out the volume of the triangular prism. Give your answer correct to the nearest whole number.
A crate is in the shape of a cuboid with inside lengths of 120 cm, 40 cm and h cm The crate has a lid.
Micah has 48 boxes. Each box is in the shape of a cube 20 cm by 20 cm by 20 cm
Micah wants to put all the boxes in the crate and shut the lid.
Work out the least possible value of h
Also involves Quadratic equations
Here is a frustum of a cone. The frustum is made by removing a small cone from a similar large cone.
The height of the large cone is 15 cm.
The radius of the base of the large cone is 6 cm. The radius of the base of the small cone is x cm. Given that the volume of the frustum is 
cm3 work out the value of x Show clear algebraic working.
Also involves Forming and solving equations
The diagram shows a rectangle and an isosceles triangle.
The perimeter of the rectangle is equal to the perimeter of the triangle.
Find the area of the rectangle.
The diagram shows a cuboid.
The volume of the cuboid is 231 cm3
Calculate the value of w
A frustum is made by removing a small square-based pyramid from a similar large squared‑based pyramid as shown in the diagram.
The height of the small pyramid is 15 cm. The height of the large pyramid is 45 cm. The square base of the large pyramid has side length 39 cm.
Work out the total surface area of the frustum. Give your answer correct to the nearest whole number.
Also involves Forming and solving equations
The diagram shows a cuboid with a square cross section.
Without using a calculator, find the 
lue of x
Also involves Converting between units
Tom puts boxes into a shipping container.
The container is a cuboid 10 metres by 2.4 metres by 2.4 metres. Each box is a cube of side 40 centimetres.
Work out the greatest number of these boxes that Tom can put into the container.
Also involves Pythagoras' theorem
The diagram shows a solid triangular prism.
Work out the total surface area of the triangular prism. Give your answer correct to 3 significant figures.
Also involves Pythagoras' theorem
The diagram shows a solid triangular prism.
Work out the total surface area of the triangular prism. Give your answer correct to 3 significant figures.
Also involves Forming and solving equations
The diagram shows a solid prism.
The cross section of the prism is shown shaded.
The volume of the prism is 924 cm3
Work out the value of x
The diagram shows two solids, A and B, made from two different metals.
Solid A is in the shape of a cylinder with radius 3 cm and height 7 cm Solid A has a mass of 2000 g
Solid B has a mass of 3375 g Solid B has a volume of 450 cm3
All of the metal from Solid A and Solid B is melted down to make a uniform Solid C
Given that there is no change to mass or volume during this process
work out the density of Solid C Give your answer correct to one decimal place.
........ g / cm3
21Given that the surface area of a sphere is 49πcm2
find the volume of the sphere. Give your answer correct to the nearest integer.
Also involves Surds · Rearranging formulae
The diagram shows a solid cone and a solid sphere.
The cone has base radius r, slant height l and perpendicular height h The sphere has radius r
The base radius of the cone is equal to the radius of the sphere.
Given that
k 
volume of the cone = volume of the sphere
show that the total surface area of the cone can be written in the form
where a is a constant to be found.
Also involves Converting between units
The diagram shows a solid cylinder made from iron.
The cylinder has diameter 18 cm and height 3.5 cm The mass of the cylinder is 7.04 kg
Work out the density of the iron. Give your answer in g/cm3 correct to 2 significant figures.
Also involves Pythagoras' theorem
A solid is made from a cone and a hemisphere.
The circular plane face of the hemisphere coincides with the circular base of the cone. The radius of the hemisphere and the radius of the circular base of the cone are both 20 cm.
The curved surface area of the cone is 580πcm2
The volume of the solid is kπcm3
Work out the exact value of k
Also involves Algebraic notation and vocabulary · Forming and solving equations
The diagram shows a frustum of a cone, and a sphere.
The frustum, shown shaded in the diagram, is made by removing the small cone from the large cone. The small cone and the large cone are similar.
The height of the small cone is hcm and the radius of the base of the small cone is rcm. The height of the large cone is kh cm and the radius of the base of the large cone is kr cm. The radius of the sphere is r cm.
The sphere is divided into two hemispheres, each of radius rcm.
Solid A is formed by joining one of the hemispheres to the frustum. The plane face of the hemisphere coincides with the upper plane face of the frustum, as shown in the diagram below.
Solid B is formed by joining the other hemisphere to the small cone that was removed from the large cone. The plane face of the hemisphere coincides with the plane face of the base of the small cone, as shown in the diagram below.
The volume of solid A is 6 times the volume of solid B.
find an expression for h in terms of k and r
The diagram shows a solid wooden cuboid.
The cuboid measures 65 cm by 35 cm by 45 cm.
A machine cuts the cuboid to make cubes. Each cube has edges of length 5 cm.
Work out the maximum number of cubes that can be made from the cuboid.
Also involves Forming and solving equations
The diagram shows a solid made from a cylinder and a hemisphere. The cylinder and the hemisphere are both made from the same metal.
The plane face of the hemisphere coincides with the upper plane face of the cylinder.
The radius of the cylinder and the radius of the hemisphere are both xcm. The height of the cylinder is 3 x cm.
The total surface area of the solid is 81πcm2 The mass of the solid is 840 grams.
The following table gives the density of each of four metals.
| Metal | Density (g/cm3) |
| Aluminium | 2.7 |
| Nickel | 8.9 |
| Gold | 19.3 |
| Silver | 10.5 |
The metal used to make the solid is one of the metals in the table.
Determine the metal used to make the solid. Show your working clearly.
Also involves Compound measures
A solid aluminium cylinder has radius 10 cm and height h cm.
The mass of the cylinder is 5.4 kg. The density of aluminium is 0.0027 kg/cm3
Calculate the value of h. Give your answer correct to one decimal place.
Also involves Rearranging formulae · Compound measures
A solid aluminium cylinder has radius 10 cm and height h cm.
The mass of the cylinder is 5.4 kg. The density of aluminium is 0.0027 kg/cm3
Calculate the value of h. Give your answer correct to one decimal place.
The diagram shows a container for water in the shape of a prism.
The rectangular base of the prism, shown shaded in the diagram, is horizontal. The container is completely full of water.
Tuah is going to use a pump to empty the water from the container so that the volume of water in the container decreases at a constant rate.
The pump starts to empty water from the container at 10 30 and at 12 00 the water level in the container has dropped by 20 cm.
Find the time at which all the water has been pumped out of the container.
Also involves Forming and solving equations
The diagram shows a cuboid and a cylinder.
The dimensions of the cuboid are xcm by 12 cm by 5 cm. The volume of the cuboid is 270 cm3
The radius of the cylinder is xcm. The height of the cylinder is 2 x cm.
Work out the volume of the cylinder. Give your answer correct to the nearest whole number.
Change 1 m3 to cm3
Also involves Ratio, proportion and rates of change
The diagram shows a container for water in the shape of a prism.
The rectangular base of the prism, shown shaded in the diagram, is horizontal. The container is completely full of water.
Tuah is going to use a pump to empty the water from the container so that the volume of water in the container decreases at a constant rate.
The pump starts to empty water from the container at 10 30 and at 12 00 the water level in the container has dropped by 20 cm.
Find the time at which all the water has been pumped out of the container.
Also involves Rounding · Forming and solving equations
The diagram shows a cuboid and a cylinder.
The dimensions of the cuboid are xcm by 12 cm by 5 cm. The volume of the cuboid is 270 cm3
The radius of the cylinder is xcm. The height of the cylinder is 2 x cm.
Work out the volume of the cylinder. Give your answer correct to the nearest whole number.
Change 1 m3 to cm3
A solid, S, is made from a hemisphere and a cylinder.
The centre of the circular face of the hemisphere and the centre of the top face of the cylinder are at the same point.
The radius of the cylinder and the radius of the hemisphere are both xcm. The height of the cylinder is (20 – 4x)cm.
Find the maximum value of y. Show clear algebraic working.
Also involves Forming and solving equations
The diagram shows a trapezium.
Work out the area of the trapezium.
The diagram shows a cuboid.
The volume of the cuboid is 806 cm3
Work out the value of x.
Write down the mathematical name of this 3-D shape.
Here is a solid cuboid.
How many faces has the cuboid?
How many vertices has the cuboid?
Work out the volume of the cuboid.
A box is to be filled with cartons.
Each carton is a cube that measures 4 cm by 4 cm by 4 cm. The box is a cuboid that measures 60 cm by 48 cm by 40 cm.
Work out the number of cartons that can completely fill the box.
The diagram shows a solid cylinder with radius 3 m.
The volume of the cylinder is 72πm3
Calculate the total surface area of the cylinder. Give your answer correct to 3 significant figures.
The diagram shows a solid cylinder with radius 3 m.
The volume of the cylinder is 72πm3
Calculate the total surface area of the cylinder. Give your answer correct to 3 significant figures.
Also involves Rounding · Converting between units
Pablo made a solid gold statue.
He melted down some gold blocks and used the gold to make the statue. Each block of gold was a cuboid, as shown below.
The mass of the statue is 5.73 kg. The density of gold is 19.32 g/cm3
Work out the least number of gold blocks Pablo melted down in order to make the statue. Show your working clearly.
Also involves Rounding
Pablo made a solid gold statue.
He melted down some gold blocks and used the gold to make the statue. Each block of gold was a cuboid, as shown below.
The mass of the statue is 5.73 kg. The density of gold is 19.32 g/cm3
Work out the least number of gold blocks Pablo melted down in order to make the statue. Show your working clearly.
Also involves Rearranging formulae
The diagram shows a solid cube. The cube is placed on a table so that the whole of one face of the cube is in contact with the table.
The cube exerts a force of 56 newtons on the table. The pressure on the table due to the cube is 0.14 newtons/cm2
Work out the volume of the cube.
Also involves Quadratic equations · Rounding
The diagram shows a cuboid of volume Vcm3
Show that V= 15 + 16x−x2− 2x3
There is a value of x for which the volume of the cuboid is a maximum.
Find this value of x. Show your working clearly. Give your answer correct to 3 significant figures.
Also involves Simplifying and manipulating expressions · Rearranging formulae
The diagram shows a frustum of a cone and a sphere.
The frustum is made by removing a small cone from a large cone. The cones are similar.
The height of the small cone is hcm. The height of the large cone is 2hcm. The radius of the base of the large cone is rcm.
The radius of the sphere is rcm.
Given that the volume of the frustum is equal to the volume of the sphere,
find an expression for r in terms of h. Give your expression in its simplest form.
Also involves Forming and solving equations
The diagram shows a prism ABCDEFGH in which ABCD is a trapezium with BC parallel to AD and CDEF is a rectangle.
BC = 7 cm AD = 12 cm DE = 10 cm
The height of trapezium ABCD is hcm The volume of the prism is 608 cm3
Work out the value of h.
The diagram shows a solid cuboid made from wood.
The wood has density 0.7 g/cm3
Work out the mass of the cuboid.
Also involves Rearranging formulae
The diagram shows a cylinder.
The cylinder has radius 8.2 cm and height 10 cm. The cylinder is empty.
Pam pours 1.5 litres of water into the cylinder.
Work out the depth of the water in the cylinder. Give your answer correct to 1 decimal place.
Also involves Forming and solving equations
A solid is made from a hemisphere and a cylinder. The plane face of the hemisphere coincides with the upper plane face of the cylinder.
The hemisphere and the cylinder have the same radius.
The ratio of the radius of the cylinder to the height of the cylinder is 1: 3
Given that the solid has volume 792
cm3 work out the height of the solid.
A cylinder has diameter 14 cm and height 20 cm.
Work out the volume of the cylinder. Give your answer correct to 3 significant figures.
A wooden box measures 30 cm by 15 cm by 32 cm. The box has a lid.
A carton measures 5 cm by 5 cm by 5 cm.
James has 110 cartons. He wants to put all these cartons in the box and be able to shut the lid.
Can James put all 110 cartons in the box and shut the lid? Show your working clearly.
Here is an empty pool in the shape of a cuboid.
The width of the pool is 3 m. The length of the pool is 12 m. The top of the pool is 2.5 m above the base of the pool.
Jeb is going to put water in the pool. The level of the surface of the water will be 60 cm below the top of the pool.
Water flows into the pool at 400 litres per minute. 1 m3 = 1000 litres
How long will it take to fill the pool to 60 cm below the top of the pool? Give your answer in hours and minutes.
Also involves Forming and solving equations · Compound measures
The diagram shows a cross section drawn through the centre, O, of the sphere.
The internal radius of the sphere is 1.2 m. The thickness of the metal is t cm.
The density of the metal is 2700 kg per m3
The mass of the 3-D shape is 1980 kg.
Work out the value of t. Give your answer correct to 2 significant figures.
The diagram shows two cylinders, A and B.
Cylinder A has height 1.6 m and radius 0.56 m.
Work out the curved surface area of cylinder A. Give your answer in m2 correct to 3 significant figures.
Cylinder B is mathematically similar to cylinder A. The height of cylinder B is 0.6 m.
Work out the radius of cylinder B.
The diagram shows two cylinders, A and B.
Cylinder A has height 1.6 m and radius 0.56 m.
Work out the curved surface area of cylinder A. Give your answer in m2 correct to 3 significant figures.
Cylinder B is mathematically similar to cylinder A. The height of cylinder B is 0.6 m.
Work out the radius of cylinder B.
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