WJEC GCSE Maths
Topic questions Geometry and measures Volume & surface area
59 questions · 281 marks · about 6 hours
Also involves Properties of triangles and quadrilaterals
Which one of the following quadrilaterals has • rotational symmetry of order 2, and • 2 lines of symmetry?
Circle the correct shape.
Square Trapezium Rhombus Kite Parallelogram
A cuboid measures 8 cm by 13 cm by 7 cm.
Calculate the volume of the cuboid.
The total surface area of all 6 faces of the cuboid below is 240 cm2.
Find the volume of the cuboid.
You must show all your working.
The diagram below shows a solid cone.
The cone has: • base radius = rcm
• total surface area = 144πcm2. Calculate the value of r.
In your working, express any areas in terms of π.
The total surface area of all 6 faces of the cuboid below is 240 cm2.
Find the volume of the cuboid.
You must show all your working.
Also involves Algebraic argument and proof
Carys says,
“It will never be possible for the volume of a cube of side length r to be equal to the volume of a hemisphere of radius r.”
Use algebraic expressions to show that Carys is correct.
The diagram shows a solid cylinder with radius 10 cm.
The total surface area of the solid cylinder is 1570·8 cm2.
Find the height of the cylinder.
The diagram shows a solid cylinder with radius 10 cm.
The total surface area of the solid cylinder is 1570·8 cm2.
Find the height of the cylinder.
The solid shown below is a quarter of a sphere with radius 7·3 cm.
Calculate the volume of this solid.
Give your answer correct to 3 significant figures.
Draw an isometric representation of a cuboid measuring 7 cm by 5 cm by 3 cm.
Use the grid below.
Calculate the volume of the cuboid.
You must give the units of your answer.
In this question, you will be assessed on the quality of your organisation, communication and accuracy in writing.
The right-angled triangle A B C is a cross-section of a prism, as shown below. A B = 9 cm, B C = 10 cm and A B C
= 90°. The length of the prism is 20 cm.
Calculate the volume of the prism. You must show all your working. [4 + 2 OCW]
Calculate the volume of the cuboid.
You must give the units of your answer.
The diagram below shows a water container.
It is formed from an open rectangular tank with a solid cylinder attached to its base inside the tank.
The base radius of the cylinder is 10 cm. The height of the cylinder is 30 cm.
How many litres of water can the container hold?
You must show all your working.
The container can hold…........ ……........ …… litres.
In this question, you will be assessed on the quality of your organisation, communication and accuracy in writing.
The right-angled triangle A B C is a cross-section of a prism, as shown below. A B = 9 cm, B C = 10 cm and A B C
= 90°. The length of the prism is 20 cm.
Calculate the volume of the prism.
You must show all your working.
The pyramid below has a horizontal rectangular base A B C D.
The point O is the top of the pyramid. It is vertically above the point A. O C = 113 cm, A B = 72 cm and C B = 84 cm.
Calculate the length of O A, the vertical height of the pyramid.
Calculate the volume of this pyramid.
Here is a net of a cuboid.
The net is folded to form a cuboid.
The volume of the cuboid is 90 cm3.
What is the value of p?
In this question, you will be assessed on the quality of your organisation, communication and accuracy in writing.
Dylan and Angharad make cuboids and cubes out of centimetre cubes.
Dylan makes the cuboid below.
Angharad makes a cube with edges of length 4 cm.
How many more centimetre cubes does Angharad use than Dylan?
You must show all your working.
Here is a net of a cuboid.
The net is folded to form a cuboid.
The volume of the cuboid is 90 cm3.
What is the value of p?
Find the volume of the solid prism shown above.
The solid prism is made of gold. Gold has a density of 19∙3 g/cm3. Calculate the mass of the prism.
Give your answer in kilograms.
Find the volume of the solid prism shown above.
The solid prism is made of gold. Gold has a density of 19∙3 g/cm3. Calculate the mass of the prism.
Give your answer in kilograms.
A sphere has a diameter of 6 cm.
A cone has a base radius of 10 cm and a height of 9 cm.
Find the ratio of the volume of the sphere to the volume of the cone.
Give your answer in its simplest form.
Volume of the sphere: Volume of the cone
A solid metal cylinder has a radius of 2∙3 cm and a height of 5 cm.
The mass of the cylinder is 423·1 g.
Find the density of the metal.
Give your answer in g/cm3.
Also involves Forming and solving equations
A cone and a cylinder have equal volumes. The cone has a base radius of rcm and a height of hcm. The cylinder has a base radius of rcm and a height of 
cm. Find h in terms of r.
You must express your answer in its simplest form.
A solid metal cylinder has a radius of 2∙3 cm and a height of 5 cm.
The mass of the cylinder is 423·1 g.
Find the density of the metal.
Give your answer in g/cm3.
A cone is joined to a hemisphere, as shown below. The cone has a base radius of 8 cm and a slant height of 17 cm. The hemisphere has the same radius as the cone.
Calculate the surface area of the composite solid.
The height of a cylinder is 24·8 cm. The ratio of the diameter of the cylinder to the height of the cylinder is 3: 2.
Find the volume of the cylinder. Give your answer correct to 2 significant figures.
You must show all your working.
Show that 5x3 
15x 
132.
A solution of the equation
lies between 2 and 3. Use the method of trial and improvement to find this solution correct to 1 decimal place.
You must show all your working.
Hence, find the height of the cuboid.
The height of a cylinder is 24·8 cm. The ratio of the diameter of the cylinder to the height of the cylinder is 3: 2.
Find the volume of the cylinder. Give your answer correct to 2 significant figures.
You must show all your working.
Show that 5x3 15x 132.
A solution of the equation
lies between 2 and 3. Use the method of trial and improvement to find this solution correct to 1 decimal place.
You must show all your working.
Hence, find the height of the cuboid.
Calculate the total surface area of a solid hemisphere with a radius of 34 cm.
A cuboid measures 5 cm by 3 cm by 2 cm.
Calculate the volume of the cuboid.
Give your answer in cm3.
A cuboid measures 5 cm by 3 cm by 2 cm.
Calculate the volume of the cuboid.
Give your answer in cm3.
Calculate the total surface area of the cuboid.
Give your answer in cm2.
A solid object is made out of a hemisphere and a cylinder.
The radius of the common circular surface is 3 cm. The volume of the whole object is 63
cm3.
Calculate the total height of the object.
Calculate the perpendicular height of a cone with a volume of 5533 cm3 and a base area of
825 cm2.
In this question, you will be assessed on the quality of your organisation, communication and accuracy in writing.
The diagrams below show a solid cone and a solid hemisphere.
The cone has a base radius of 9 cm and a vertical height of 12 cm. The hemisphere has a radius of 8 cm.
Which of the two solids has the greater curved surface area? You should express any areas in terms of 
.
You must show all your working.
A solid object is made by drilling a cylindrical hole of radius 4 cm through a cuboid measuring
20 cm by 15 cm by 10 cm as shown below.
Calculate the volume of the object.
Give your answer in cm3.
The object is made from a material which has a density of 2·4 g/cm3. Calculate the mass of the object.
Give your answer in kg, correct to the nearest kg.
In this question, you will be assessed on the quality of your organisation, communication and accuracy in writing.
Cuboid A is made up of a number of cubes as shown below. Each edge of each cube is 1 cm long.
A different cuboid, Cuboid B, has the same length and width as Cuboid A. The height of Cuboid B is three times the height of Cuboid A.
What is the volume of Cuboid B?
You must show all your working.
Also involves Rearranging formulae
The diagram shows a cylinder.
The cylinder has a base of radius r and a height of 
A sphere has radius R. The volume of the sphere is equal to the volume of the cylinder. Find R in terms of r.
Give your answer in its simplest form.
A cuboid has a volume of 97.6 cm3. 10. The length of the cuboid is 6.1 cm and its width is 5 cm.
Calculate the height of the cuboid.
The diagram below shows an empty cylinder, with radius 10 cm and height 20 cm.
Using 
= 3.14, calculate the volume of the cylinder. (a)
What is the greatest whole number of litres that this cylinder can hold?
A cuboid has sides x cm, 5 cm and 7 cm.
The total surface area of the cuboid is 142 cm2.
Form an equation in terms of x.
Solve the equation to find x.
The diagram below shows an empty cylinder, with radius 10 cm and height 20 cm.
Using 
= 3.14, calculate the volume of the cylinder. (a)
What is the greatest whole number of litres that this cylinder can hold?
A cuboid has sides x cm, 5 cm and 7 cm.
The total surface area of the cuboid is 142 cm2.
Form an equation in terms of x.
Solve the equation to find x.
Calculate the total surface area of a solid hemisphere with a base radius of 29 cm.
Cube A and cuboid B are shown below.
Express the volume of B as a percentage of the volume of A.
You must show all your working.
Draw an isometric representation of a cuboid measuring 6 cm by 4 cm by 3 cm.
Use the grid below.
Calculate the volume of the cuboid.
Give the units of your answer.
In this question, you will be assessed on the quality of your organisation, communication and accuracy in writing.
Cube A and cuboid B are shown below.
Express the volume of B as a percentage of the volume of A.
You must show all your working.
Calculate the volume of a pyramid with a base area of 13 200 cm2 and a perpendicular height of 460 cm.
Give your answer in m3.
How many cubic centimetres (cm3) are there in 2·5 litres?
2.5 litres =........ cm3
Also involves Simplifying and manipulating expressions
A cylinder just fits inside a hollow cube with sides of length m cm.
The radius of the cylinder is 
cm. The height of the cylinder is mcm.
The ratio of the volume of the cube to the volume of the cylinder is given by
volume of cube: volume of cylinder
= k: 
,
where k is a number.
Find the value of k.
You must show all your working.
A triangular prism of length 2 metres is shown below.
AC = 21 cm, BC = 35 cm and BAC
= 90°.
Calculate the area of triangle ABC. Give your answer in cm2.
You must show all your working.
Calculate the volume of the prism.
You must give the units of your answer.
Also involves Simplifying and manipulating expressions
A cylinder just fits inside a hollow cube with sides of length m cm.
The radius of the cylinder is cm. The height of the cylinder is mcm.
The ratio of the volume of the cube to the volume of the cylinder is given by
volume of cube: volume of cylinder
= k: ,
where k is a number.
Find the value of k.
You must show all your working.
A triangular prism of length 2 metres is shown below.
AC = 21 cm, BC = 35 cm and BAC
= 90°.
In this part of the question, you will be assessed on the quality of your organisation, communication and accuracy in writing. Calculate the area of triangle ABC. Give your answer in cm2.
You must show all your working.
Calculate the volume of the prism.
You must give the units of your answer.
A cone is joined to a cylinder, as shown below.
The cone has a base radius of 11 cm and a slant height of 13 cm. The cylinder has the same radius, 11 cm, and a height of 17 cm.
Calculate the total surface area of the composite solid.
By considering algebraic expressions, show that it will never be possible for the surface area of
a sphere of radius r to be equal to the surface area of a cube with sides of length r.
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