Volume & Surface Area (OCR GCSE Maths)

Here is a plan of Martin's driveway.

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.

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.

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

 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.

Here is a solid prism.

Work out the volume of the prism.

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

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.

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

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.

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

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.

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 .

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?

The diagram shows a solid hemisphere of radius 5 cm.

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

The diagram shows a solid shape.

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 cm²

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

Here is a solid square-based pyramid, VABCD.

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.

Work out the total surface area of the pyramid.

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.

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

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 , where  is an integer.

Work out the value of .

A container is in the shape of a cuboid.

The container is full of water.

A cup holds 275 m of water.

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

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

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

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

The diagram shows a solid metal cylinder.

The cylinder has base radius  and height .

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

Find an expression for  in terms of .

The diagram shows a pyramid.

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.

Find the size of angle DAB.

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.

The diagram shows prism.

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

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

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

The volume of S is 576 cm³.

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

 is a solid pyramid.

is a square of side 20 cm.

The angle between any sloping edge and the plane  is .

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

The diagram shows a solid hemisphere.

The volume of the hemisphere is 
Work out the exact total surface area of the solid hemisphere.
Give your answer as a multiple of .

The diagram shows a cuboid .

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

The diagram shows a solid shape.

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 cm³.

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

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

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)

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

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

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

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)

Here are four graphs.

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.

Sumeet has a pond in the shape of a prism.

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.

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

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

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  area of base × perpendicular height]

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

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

Calculate the total surface area of the cube.

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

A cone has radius r cm and height h cm.

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 ]

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

The diagram shows a cylinder and a sphere.

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

Work out the radius  cm of the sphere.

[The volume  of a sphere with radius  is .]

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

The diagram shows a container for grain.

The container is a cylinder on top of a cone.
The cylinder has a radius of 3 m and a height of 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 .
Give your answer as a fraction in its simplest form.
You must show all your working.

The diagram shows a sphere and a solid cylinder.

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

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

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)

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

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.

The diagram shows a solid cone.

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

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

Find the value of .

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

The total height of the solid is 
The radius of the base of the cone is 
The radius of the hemisphere is 

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

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

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

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

Calculate the value of .
Show your working.

[The volume of a cone with radius r and height is ]

............................

The diagram shows a cylinder and a cone.

The cylinder has radius 2cm and height 9cm.
The cone has radius cm and height cm.

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

Work out the value of .

[The volume of a cone with radius and height is ]

Calculate the volume of a sphere with radius 6cm.

[The volume of a sphere with radius is . ]

...................... cm³

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%.

Calculate the perpendicular height of the pyramid.

[The volume of a pyramid is area of base  perpendicular height.

A hemisphere is half a sphere.]

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

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

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  × area of base × perpendicular height.]

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

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 .]

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.

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

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

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

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

Here is a sector, , of a circle with centre and angle = 

The sector can form the curved surface of a cone by joining to .

The height of the cone is .
The volume of the cone is 

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

.......................

The diagram shows a solid prism .

The trapezium , in which is parallel to , is a cross section of the prism.
The base of the prism is a horizontal plane.
and are rectangles.
The midpoint of is vertically above the midpoint of so that .

The perpendicular distance between edges and is 20 .

Work out the total surface area of the prism.

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 
The density of gold is 

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

Here is a tunnel for a toy train.

The diagram below shows the cross section of the tunnel.

 is a semicircular arc of radius 10 cm
 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 .

....................cm²

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

Volume of a sphere =where is the radius

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

Right-angled triangleis the cross section of a prism.

is the midpoint of .

Work out the volume of the prism.

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

A solid shape is made by joining two cones.

Each cone has the same radius.

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 cm²

Curved surface area of a cone = .  where is the radius and is the slant height

Work out the radius.