Volume & Surface Area

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3 marks

The diagram shows a prism.

q1-4-6-easy-edexcel-gcse-maths

Work out the volume of the prism.

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4 marks

Here is a triangular prism.

j80Xn1dy_q2-4-6-easy-edexcel-gcse-maths

Work out the volume of this triangular prism.

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3 marks

The diagram shows a prism.

h61o9nj~_q3-4-6-easy-edexcel-gcse-maths

The area of the cross section of the prism is 30 cm².
The length of the prism is 25 cm.

Work out the volume of the prism.

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3 marks

Here is a cuboid.

KbQWrINb_q4-4-6-easy-edexcel-gcse-maths

The cuboid is 6 cm by 1.5 cm by 1.5 cm.

Work out the total surface area of the cuboid.

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1 mark

Circle the volume, in cm³ , of a cylinder with radius 5 cm and height 8 cm.

40 π

80 π

200 π

1600 π

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4 marks

Here are two solids.

q9-paper-1h-nov-2019-aqa-gcse-maths

q9-2-paper-1h-nov-2019-aqa-gcse-maths

Which solid has the greater volume?
You must show your working.

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2 marks

Ashraf is going to put boxes into a crate.

The crate is a cuboid measuring 2.5 m by 2 m by 1.2 m
Each box is a cube of length 50 cm

He does these calculations.

\begin{array}{rcl} volume of crate & = & 2.5 \times 2 \times 1.2 \\ = & 6 m^{3} \\ volume of one box & = & 0.5 \times 0.5 \times 0.5 \\ = & 0.125 m^{3} \\ number of boxes & = & 6 \div 0.125 \\ = & 48 \end{array}

He claims,

‘‘I can put 48 boxes in the crate.’’

Evaluate Ashraf’s method and claim.

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2 marks

3.8 m³ of concrete is made into the shape of a cylinder.

The base has radius 0.5 metres.

q16b-paper-2h-june-2018-aqa-gcse-maths

Work out the height of the cylinder.

................................m

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1 mark

The graph shows information about prisms with the same volume.

q6-paper2h-nov2017-aqa-gcse-maths

Give one example to show the volume is 24 cm³.

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2 marks

Steph is solving a problem.
Cube A has a surface area of 150 cm²
Cube B has sides half the length of cube A

What is the volume of cube B?

To solve this problem, Steph decides to

halve the surface area
calculate the square root of the answer
then divide by 6
then cube this answer to work out the volume.

Evaluate Steph’s method.

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3 marks

The diagram shows a cuboid and a cylinder.

q5-4ma1-1hr-qp-jan21-paper1-igcse-maths

The dimensions of the cuboid are $x$ cm by 12 cm by 5 cm.
The volume of the cuboid is 270 cm³

The radius of the cylinder is $x$ cm.
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.

...................................................... cm³

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2 marks

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.

......................... ${cm}^{3}$

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2 marks

A cylinder has height 1.6 m and radius 0.56 m.
Work out the curved surface area of the cylinder.
Give your answer in m² correct to 3 significant figures.

....................................................... m²

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5 marks

Here is a triangular prism.

q10-4ma1-2h-qp-june21-paper2-igcse-maths

Work out the volume of the prism.
Give your answer correct to 3 significant figures.

....................................................... cm³

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2 marks

The diagram shows a right-angled triangular prism ABCDEF.

q18-paper4-nov2019-ocr-gcse-maths

Length AD = 11 cm, length CD = 10 cm and length CF = 6 cm.

Calculate the volume of the prism.

................................................... cm³

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3 marks

The following formula is for the area, $A$ , of the curved surface area of a cone.
$A = π r l$ , where $r$ is the radius and $l$ is the slant height of the cone.

Calculate the total surface area of a cone with radius 5cm and slant height 12cm.

................ cm²

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17a

2 marks

A circular table top has radius 70 cm.

Calculate the area of the table top in cm², giving your answer as a multiple of $π$ .

....................... cm²

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17b

2 marks

The volume of the table top is 17 150 $π$ cm³.

Calculate the thickness of the table top.

........................ cm

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3 marks

The diagram shows a solid metal cylinder.

q1-4-7-hard-edexcel-gcse-maths

The cylinder has base radius $2 x$ and height $9 x$ .

The cylinder is melted down and made into a sphere of radius $r$ .

Find an expression for $r$ in terms of $x$ .

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4 marks

The diagram shows a pyramid.

screenshot-2023-05-26-at-2-17-38-pm

BCDE is a square with sides of length 10 cm,
The other faces of the pyramid are equilateral triangles with sides of length 10 cm.
Calculate the volume of the pyramid.
Give your answer correct to 3 significant figures.

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2 marks

Find the size of angle DAB.

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4 marks

q2-4-7-hard-edexcel-gcse-maths

A frustrum is made by removing a small cone from a similar large cone.

The height of the small cone is 20 cm.
The height of the large cone is 40 cm.
The diameter of the base of the large cone is 30 cm.

Work out the volume of the frustrum.
Give your answer correct to 3 significant figures.

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4 marks

The diagram shows prism.

q4-4-7-hard-edexcel-gcse-maths

All measurements are in centimetres.
All corners are right angles.

Find an expression, in terms of $x$ , for the volume, in cm³, of the prism.
You must show your working.
Give your answer in its simplest form.

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5 marks

Shape S is one quarter of a solid sphere, centre $O$ .

q5-4-7-hard-edexcel-gcse-maths

The volume of S is 576 $π$ cm³.

Find the surface area of S.
Give your answer correct to 3 significant figures.
You must show your working.

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5 marks

$V A B C D$ is a solid pyramid.

q6-4-7-hard-edexcel-gcse-maths

$A B C D$ is a square of side 20 cm.

The angle between any sloping edge and the plane $A B C D$ is $55 °$ .

Calculate the surface area of the pyramid.
Give your answer correct to 2 significant figures.

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4 marks

The diagram shows a solid hemisphere.

q7-4-7-hard-edexcel-gcse-maths

The volume of the hemisphere is $\frac{250}{3} π$
Work out the exact total surface area of the solid hemisphere.
Give your answer as a multiple of $π$ .

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4 marks

The diagram shows a cuboid $A B C D E F G H$ .

q8-4-7-hard-edexcel-gcse-maths

$A B = 7 cm, A F = 5 cm and F C = 15 cm .$

Calculate the volume of the cuboid.
Give your answer correct to $3$ significant figures.

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5 marks

The diagram shows a solid shape.

q9-4-7-hard-edexcel-gcse-maths

The solid shape is made from a hemisphere and a cone.
The radius of the hemisphere is equal to the radius of the base of the cone.

The cone has a height of 10 cm.
The volume of the cone is 270 $π$ cm³.

Work out the total volume of the solid shape.
Give your answer in terms of $π$ .

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10a

5 marks

The diagram shows an oil tan in the shape of a prism.
The cross section of the prism is a trapezium.

screenshot-2023-05-26-at-2-22-08-pm

The tank is empty.

Oil flows into the tank.
After one minute there are 300 litres of oil in the tank.

Assume that oil continues to flow into the tank at this rate.

Work out how many more minutes it takes for the tank to be 85% full of oil.
(1m³ = 1000 litres)

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10b

1 mark

The assumption about the rate of flow of the oil could be wrong.

Explain how this could affect your answer to part (a).

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11a

5 marks

The diagram shows a swimming pool in the shape of a prism.

screenshot-2023-05-26-at-2-26-17-pm

The swimming pool is empty.

The swimming pool is filled with water at a constant rate of 50 litres per minute.

Work out how long it will take for the swimming pool to be completely full of water.
Give your answer in hours.
(1 m³= 1000 litres)

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11b

1 mark

Here are four graphs.

screenshot-2023-05-26-at-2-29-08-pm

Write down the letter of the graph that best shows how the depth of the water in the pool above the line MN changes with time as the pool is filled.

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6 marks

Sumeet has a pond in the shape of a prism.

q10-4-7-hard-edexcel-gcse-maths

The pond is completely full of water.
Sumeet wants to empty the pond so he can clean it.
Sumeet uses a pump to empty the pond.

The volume of water in the pond decreases at a constant rate.
The level of the water in the pond goes down by 20 cm in the first 30 minutes.

Work out how much more time Sumeet has to wait for the pump to empty the pond completely.

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4 marks

A cuboid measures 6 cm by 8 cm by 15 cm.
A cube has the same volume as the cuboid.

Find the surface area of the cube, giving your answer correct to 3 significant figures.

.................................................... cm²

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4 marks

An octahedron is formed from two identical square based pyramids.
The square bases are stuck together as shown.

q8-paper6-nov2019-ocr-gcse-maths

The volume of the octahedron is 60 cm³.
The length of the side of each pyramid’s square base is 5 cm.

Work out the height h cm of the octahedron.
[The volume of a pyramid is $\frac{1}{3} \times$ area of base × perpendicular height]

h = ............................................... cm

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5 marks

The length of the longest diagonal of a cube is 25cm.

Calculate the total surface area of the cube.

................................................... cm²

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4 marks

A cone has radius r cm and height h cm.

q18-paper6-june2019-ocr-gcse-maths

The height is three times the radius.
The volume of the cone is 2100 cm³.

Calculate the radius of the cone.

[The volume V of a cone with radius r and height h is $V = \frac{1}{3} π r^{2} h .$ ]

..................................................... cm

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5 marks

The diagram shows a cylinder and a sphere.

q6-paper4-nov2018-ocr-gcse-maths

The cylinder has radius 12cm and height 30cm.
The cylinder and the sphere have the same volume.

Work out the radius $r$ cm of the sphere.

[The volume $V$ of a sphere with radius $r$ is $V = \frac{4}{3} π r^{3}$ .]

....................... cm

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5 marks

Here is a plan of Martin's driveway.

q1-4-7-medium-edexcel-gcse-maths

Martin is going to cover his driveway with gravel.
The gravel will be 6 cm deep.

Gravel is sold in bags.
There are 0.4 m³ of gravel in each bag.
Each bag of gravel costs £38

Martin gets a discount of 30% off the cost of the gravel.

Work out the total amount of money Martin pays for the gravel.

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4 marks

The diagram shows a container for oil.
The container is in the shape of a cuboid.
The container is empty.

Sally has to fill the container with oil.
A bottle of oil costs £3.50
There are 3000 cm³ of oil in each bottle.

Sally must not spend more than £60 buying the oil.

Can Sally buy enough oil to fill the container?
You must show all your working.

q2-4-7-medium-edexcel-gcse-maths

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3 marks

Jane has a carton of orange juice.
The carton is in the shape of a cuboid.

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The depth of the orange juice in the carton is 8 cm.

Jane closes the carton.
Then she turns the carton over so that it stands on the shaded face.

Work out the depth, in cm, of the orange juice now.

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3 marks

Here is a solid prism.

q4-4-7-medium-edexcel-gcse-maths

Work out the volume of the prism.

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3 marks

The diagram shows a large tin of pet food in the shape of a cylinder.

q5-4-7-medium-edexcel-gcse-maths

The large tin has a radius of 6.5 cm and a height of 11.5 cm.

A pet food company wants to make a new size of tin.

The new tin will have a radius of 5.8 cm.
It will have the same volume as the large tin.

Calculate the height of the new tin.
Give your answer correct to one decimal place.

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3 marks

The diagram shows a solid made from a hemisphere and a cone.

q6-4-7-medium-edexcel-gcse-maths

The radius of the hemisphere is 4 cm.
The radius of the base of the cone is 4 cm.

Calculate the volume of the solid.
Give your answer correct to 3 significant figures.

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3 marks

Jane makes cheese.
The cheese is in the shape of a cuboid.

q7-1-4-7-medium-edexcel-gcse-maths

Jane is going to make a new cheese.

The new cheese will also be in the shape of a cuboid.
The cross section of the cuboid will be a 5 cm by 5 cm square.

q7-2-4-7-medium-edexcel-gcse-maths

Jane wants the new cuboid to have the same volume as the 2 cm by 10 cm by 15 cm cuboid.

Work out the value of $x$ .

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4 marks

Anne wants to fill 12 hanging baskets with compost.

Each hanging basket is a hemisphere of diameter 40 cm.

Anne has 4 bags of compost.
There are 50 litres of compost in each bag.

Has Anne got enough compost to fill the 12 hanging baskets?

q8-4-7-medium-edexcel-gcse-maths

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3 marks

The diagram shows a solid hemisphere of radius 5 cm.

q9-4-7-medium-edexcel-gcse-maths

Find the total surface area of the solid hemisphere.
Give your answer in terms of $π$ .

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5 marks

The diagram shows a solid shape.

q10-4-7-medium-edexcel-gcse-maths

The solid shape is made from a cylinder and a hemisphere.
The radius of the cylinder is equal to the radius of the hemisphere.

The cylinder has a height of 10 cm.
The curved surface area of the hemisphere is 32 $π$ cm².

Work out the total surface area of the solid shape.
Give your answer in terms of $π$ .

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11a

2 marks

Here is a solid square-based pyramid, VABCD.

screenshot-2023-05-26-at-2-08-21-pm

The base of the pyramid is a square of side 6 cm.

The height of the pyramid is 4 cm.

M is the midpoint of BC and VM = 5 cm.

Draw an accurate front elevation of the pyramid from the direction of the arrow.

screenshot-2023-05-26-at-2-11-12-pm

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11b

4 marks

Work out the total surface area of the pyramid.

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5 marks

The diagram shows a sand pit.
The sand pit is in the shape of a cuboid.

Sally wants to fill the sand pit with sand.
A bag of sand costs £2.50
There are 8 litres of sand in each bag.

Sally says,

"The sand will cost less than £70"

Show that Sally is wrong.

q12-4-7-medium-edexcel-gcse-maths

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4 marks

The diagram shows a solid shape.
The shape is a cone on top of a hemisphere.

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The height of the cone is 10 cm.
The base of the cone has a diameter of 6 cm.
The hemisphere has a diameter of 6 cm.

The total volume of the shape is $k {πcm}^{3}$ , where $k$ is an integer.

Work out the value of $k$ .

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4 marks

A container is in the shape of a cuboid.

q14-4-7-medium-edexcel-gcse-maths

The container is $\frac{2}{3}$ full of water.

A cup holds 275 m $l$ of water.

What is the greatest number of cups that can be completely filled with water from the container?

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5 marks

Jeremy has to cover 3 tanks completely with paint.

Each tank is in the shape of a cylinder with a top and a bottom.
The tank has a diameter of 1.6 m and a height of 1.8 m.

Jeremy has 7 tins of paint.
Each tin of paint covers 5 m²

Has Jeremy got enough paint to cover completely the 3 tanks?
You must show how you get your answer.

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3 marks

Frances grows plants in a container.
Each of the 5 faces of the container is made of glass.

q16-4-7-medium-edexcel-gcse-maths

The container is in the shape of a prism.
The cross section of the prism is an isosceles triangle with height $40 cm$ .

$B C = 60 cm A B = A C = 50 cm C P = 80 cm$

Work out the total area of glass needed to make the container.

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5 marks

The diagram shows a container for grain.

q1-4-7-very-hard-edexcel-gcse-maths

The container is a cylinder on top of a cone.
The cylinder has a radius of 3 m and a height of $h$ m.
The cone has a base radius of 3 m and a vertical height of 4 m.

The container is empty.
The container is then filled with grain at a constant rate.

After 5 hours the depth of the grain is 6 metres above the vertex of the cone.
After 9 hours the container is full of grain.

Work out the value of $h$ .
Give your answer as a fraction in its simplest form.
You must show all your working.

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5 marks

The diagram shows a sphere and a solid cylinder.

q2-4-7-very-hard-edexcel-gcse-maths

The sphere has radius 6 cm.
The solid cylinder has a base radius of 3 cm and a height of $h$ cm.

The total surface area of the cylinder is twice the total surface area of the sphere.

Work out the ratio of the volume of the sphere to the volume of the cylinder.
Give your answer in its simplest form.
You must show all your working.

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5 marks

The diagram shows a swimming pool in the shape of a prism.

q3-4-7-very-hard-edexcel-gcse-maths

The swimming pool is empty.

Water from 3 water tankers is going to be put into the pool.
There are 20 000 litres of water in each water tanker.

Sam thinks that the surface of the water in the pool will be 10 cm below the top of the pool.

Is Sam correct?
You must show how you get your answer.
(1 m³ = 1000 litres)

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5 marks

A frustum is made by removing a small cone from a large cone as shown in the diagram.

q4-4-7-very-hard-edexcel-gcse-maths

The frustum is made from glass.
The glass has a density of 2.5 g / cm³

Work out the mass of the frustum.
Give your answer to an appropriate degree of accuracy.

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5 marks

The diagram shows a solid cone.

q5-4-7-very-hard-edexcel-gcse-maths

The diameter of the base of the cone is 24 $x$ cm.
The height of the cone is 16 $x$ cm.

The curved surface area of the cone is 2160 $π$ cm².
The volume of the cone is $V π$ cm³, where $V$ is an integer.

Find the value of $V$ .

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5 marks

A solid is made by putting a hemisphere on top of a cone.

q6-1-4-7-very-hard-edexcel-gcse-maths

The total height of the solid is $5 x .$
The radius of the base of the cone is $x .$
The radius of the hemisphere is $x .$

q6-2-4-7-very-hard-edexcel-gcse-maths

A cylinder has the same volume as the solid.
The cylinder has radius $2 x$ and height $h .$
All measurements are in centimetres.

Find a formula for $h$ in terms of $x .$
Give your answer in its simplest form.

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5 marks

Here is a tunnel for a toy train.

q18-1-paper-1h-june-2018-aqa-gcse-maths

The diagram below shows the cross section of the tunnel.

q18-2-paper-1h-june-2018-aqa-gcse-maths

$A D$ is a semicircular arc of radius 10 cm
$B C$ is a semicircular arc of radius 7 cm
The length of the tunnel is 30 cm

Work out the total area of all six faces of the tunnel.
Give your answer in terms of $π$ .

....................cm²

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4 marks

A ball contains 5000 cm³ of air.

More air is pumped into the ball at a rate of 160 cm³ per second.
The ball is full of air when it becomes a sphere with radius 15 cm

q9-paper-2h-nov-2021-aqa-gcse-maths

Volume of a sphere =

\frac{4}{3} π r^{3}

where

r

is the radius

Does it take less than 1 minute to fill the ball?
You must show your working.

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4 marks

Right-angled triangle $A B C$ is the cross section of a prism.

$A B = 5 cm B C = 12 cm$
$M$ is the midpoint of $C F$ .
$A M = 16 cm$

q23-paper-2h-nov-2020-aqa-gcse-maths

Work out the volume of the prism.

......................................cm³

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3 marks

A solid shape is made by joining two cones.

Each cone has the same radius.

q21-paper-2h-june-2019-aqa-gcse-maths

One cone has slant height = 2 × radius
The other cone has slant height = 3 × radius
The total surface area of the shape is 57.8 $π$ cm²

Curved surface area of a cone =

π r l

. where

r

is the radius and

l

is the slant height

Work out the radius.

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4 marks

A frustum is made by removing a small cone from a large cone.
The cones are mathematically similar.

q16-paper1h-june2018-edexcel-igcse-maths

The large cone has base radius $r$ cm and height $h$ cm.
Given that

$\frac{volume of frustum}{volume of large cone} = \frac{98}{125}$

find an expression, in terms of $h$ , for the height of the frustum.

....................................................... cm

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6 marks

Here is a sector, $A O B$ , of a circle with centre $O$ and angle $A O B$ = $x °$

q26-4ma1-2h-qp-jan20-paper2-igcse-maths

The sector can form the curved surface of a cone by joining $O A$ to $O B$ .

q26-14ma1-2h-qp-jan20-paper2-igcse-maths

The height of the cone is $25 cm$ .
The volume of the cone is $1600 {cm}^{3}$

Work out the value of $x .$
Give your answer correct to the nearest whole number.

$x =$ .......................

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4 marks

The diagram shows a solid prism $A B C D E F G H$ .

q17-4ma1-2h-qp-jan21-paper2-igcse-maths

The trapezium $A B C D$ , in which $A D$ is parallel to $B C$ , is a cross section of the prism.
The base $A D E H$ of the prism is a horizontal plane.
$A D E H$ and $B C F G$ are rectangles.
The midpoint of $B C$ is vertically above the midpoint of $A D$ so that $B A = C D$ .

$A D = 37 cm$ $G F = 28 cm$ $D E = 24 cm$

The perpendicular distance between edges $A D$ and $B C$ is 20 $cm$ .

Work out the total surface area of the prism.

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5 marks

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.

q9-4ma1-1hr-qp-jan20-paper1-igcse-maths

The mass of the statue is $5.73 kg .$
The density of gold is $19.32 g / {cm}^{3}$

Work out the least number of gold blocks Pablo melted down in order to make the statue.
Show your working clearly.

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5 marks

The base of a cone is fixed to the top of a cylinder to make a decoration.

q14-paper6-nov2020-ocr-gcse-maths

The radius of the base of the cone and of the cylinder is $r$ cm.
The cone’s height is 5 $r$ cm.
The total height of the decoration is 6 $r$ cm.
The total volume of the decoration is 225 cm³.

Calculate the value of $r$ .
Show your working.

[The volume $V$ of a cone with radius r and height $h$ is $V = \frac{1}{3} π r^{2} h$ ]

$r =$ ............................

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5 marks

The diagram shows a cylinder and a cone.

q14-paper5-june2018-ocr-gcse-maths The cylinder has radius 2cm and height 9cm.
The cone has radius $r$ cm and height $h$ cm.

The ratio $r : h$ is 1 : 4.
The volume of the cone is equal to the volume of the cylinder.

Work out the value of $r$ .

[The volume $V$ of a cone with radius $r$ and height $h$ is $V = \frac{1}{3} π r^{2} h$ ].

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17a

2 marks

Calculate the volume of a sphere with radius 6cm.

q13a-paper6-june2018-ocr-gcse-maths

[The volume $V$ of a sphere with radius $r$ is $V = \frac{4}{3} π r^{3}$ . ]

...................... cm³

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17b

5 marks

An ornament is made from a solid glass square-based pyramid.
The base has side length 15cm.
A hemisphere with radius 6cm is cut out of the base of the pyramid.
This reduces the volume of glass contained in the ornament by 30%.

q13b-paper6-june2018-ocr-gcse-maths

Calculate the perpendicular height of the pyramid.

[The volume of a pyramid is $\frac{1}{3} \times$ area of base $\times$ perpendicular height.

A hemisphere is half a sphere.]

....................... cm

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5 marks

The object below is made from a square-based pyramid joined to a cuboid.

q8-paper6-june2017-ocr-gcse-maths

The base of the cuboid and the base of the pyramid are both squares of side 4 cm.
The height of the cuboid is 8cm and the total height of the object is 13cm.
The total mass of the object is 158 g.

The cuboid is made from wood with density 0.67 g/cm³.
The pyramid is made from granite.

Calculate the density of the granite.

[The volume of a pyramid is $\frac{1}{3}$ × area of base × perpendicular height.]

.................................................g/cm³

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4 marks

The volume of Earth is 1.08 × 10¹² km³.
The volume of Jupiter is 1.43 × 10¹⁵ km³.

How many times larger is the radius of Jupiter than the radius of Earth?
Assume that Jupiter and Earth are both spheres.

[The volume v of a sphere with radius r is $v = \frac{4}{3} π r^{3}$ .]

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4 marks

A sculptor needs to lift a piece of marble.
It is a cuboid with dimensions 1m by 0.5m by 0.2m.
Marble has a density of 2.7g/cm³.
The sculptor’s lifting gear can lift a maximum load of 300kg.

Can the lifting gear be used to lift the marble?
Justify your decision.

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AQA GCSE Maths

Topic Questions

Volume & Surface Area