Volume & Surface Area (CIE IGCSE Maths: Extended)

A pipe is completely full of water.
Water flows through the pipe at a speed of 1.2 m/s into a tank.
The cross-section of the pipe has an area of 6 cm².

Calculate the number of litres of water flowing into the tank in 1 hour.

.......................................... litres

A cone with height 14.8 cm has volume 275 cm³.
 
Calculate the radius of the cone.
[The volume, , of a cone with radius  and height  is .]
 

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

A pipe is full of water.
The cross-section of the pipe is a circle, radius 2.6 cm.
Water flows through the pipe into a tank at a speed of 12 centimetres per second.
 
Calculate the number of litres that flow into the tank in one hour.
 

............................................ litres

The diagram shows a sector of a circle with centre , radius 8 cm and sector angle 165°.
The surface area of a sphere is the same as the area of the sector.

Calculate the radius of the sphere.
[The surface area, , of a sphere with radius is  ]

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

A cone is made from the sector by joining to .

Show that the volume of a metal sphere of radius 15 cm is 14 140 cm³, correct to 4 significant figures.
[The volume, , of a sphere with radius  is .]

A solid metal sphere with radius 6 cm is melted down and all of the metal is used to make a solid cone with radius 8 cm and height cm.

Show that .
[The volume, , of a sphere with radius is .]
[The volume, , of a cone with radius and height is .]

[2]

.............................................. cm [2]

Calculate the curved surface area of the cone.
[The curved surface area, , of a cone with radius and slant height is .]

............................................. cm² [1]

Work out the volume of the prism.
 

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

The diagram shows a solid metal prism.
The volume of the prism is 2187 cm³.
The larger prism is melted down into a sphere.
 
Calculate the radius of the sphere.
[The volume, , of a sphere with radius is .]
 

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

 
The diagram shows the surface of a garden pond, made from a rectangle and two semicircles.
The rectangle measures 3 m by 1.2 m.

Calculate the area of this surface.
 

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

The pond is a prism and the water in the pond has a depth of 20 cm.

Calculate the number of litres of water in the pond.
 

........................................... litres

After a rainfall, the number of litres of water in the pond is 1007.

Calculate the increase in the depth of water in the pond.
Give your answer in centimetres.
 

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

A sphere has radius cm. The volume of the sphere is 1000 cm³.
 
Calculate the value of .
[The volume, , of a sphere with radius  is .] 
 

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

The diagram shows a prism with a right-angled triangle as its cross-section. The volume of the prism is 1000 cm³.
 
Calculate the value of .
 

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

A gold model is made.
This model is a prism with a cross-section of area 77.44 cm².
 
This gold model is 15 mm thick.
One cubic centimetre of gold has a mass of 19 grams.
 
Calculate the mass of the gold model in kilograms.
 

.............................................. kg

The diagram shows a prism with length 18 cm and volume 253.8 cm³.
The cross-section of the prism is a right-angled triangle with base 6 cm and height  cm.

Show that the value of  is 4.7 .

Calculate the total surface area of the prism.
 

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

The diagram shows a solid cone.
The radius is 8 cm and the slant height is 17 cm.

Calculate the curved surface area of the cone.
[The curved surface area, , of a cone with radius and slant height  is .]
 

........................................... cm² [2]

Calculate the volume of the cone.
[The volume, , of a cone with radius  and height is .]

........................................... cm³ [4]

............................................... g [1]

............................................... g [1]

A solid hemisphere has volume 230 cm³.

Calculate the radius of the hemisphere.
[The volume, , of a sphere with radius is .]

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

A solid cylinder with radius 1.6 cm is attached to the hemisphere to make a toy.

The total volume of the toy is 300 cm³.

Calculate the height of the cylinder.

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

Water flows at a speed of 20 cm/s along a rectangular channel into a lake.
The width of the channel is 15 cm.
The depth of the water is 2.5 cm.
 
Calculate the amount of water that flows from the channel into the lake in 1 hour.
Give your answer in litres.
 

 ........................................ litres

Water from Manjeet’s shower flows at a rate of 12 litres per minute.
The water from the shower flows into a tank that is a cuboid of length 90 cm and width 75 cm.

Calculate the increase in the level of water in the tank when the shower is used for 7 minutes.

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

............................................... m² [2]

$ ................................................... [2]

The lake behind a dam has an area of 55 hectares.
When the gates in the dam are open, water flows out at a rate of 75 000 litres per second.

[1]

.......................................... cm [3]

Water flows through a cylindrical pipe at a speed of 8 cm/s.
The radius of the circular cross-section is 1.5 cm and the pipe is always completely full of water.

Calculate the amount of water that flows through the pipe in 1 hour.
Give your answer in litres.

....................................... litres

A cylinder with radius 6 cm and height  cm has the same volume as a sphere with radius 4.5 cm.
 
Find the value of .
[The volume, , of a sphere with radius  is .]
 

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

A solid metal cube of side 20 cm is melted down and made into 40 solid spheres, each of radius  cm.
 
Find the value of .
[The volume, , of a sphere with radius  is .]
 

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

A solid metal cone has radius 1.65 cm and slant height 4.70 cm.

Calculate the total surface area of the cone.
[The curved surface area, , of a cone with radius  and slant height  is .]

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

Calculate the volume of the cone.
[The volume, , of a cone with radius  and height  is .]

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

A metal sphere with radius 5 cm is melted down to make cones identical to this one.

Calculate the number of complete identical cones that are made.

[The volume, , of a sphere with radius  is .]

A cone has radius cm and slant height  cm. The volume of the cone is 1000 cm³.
 
Calculate the value of .
[The volume, , of a cone with radius  and height  is .] 
 

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

The volume of a solid metal sphere is .

Calculate the radius of the sphere.
[The volume, , of a sphere with radius  is .] 
 

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

The metal sphere is placed in an empty tank.
The tank is a cylinder with radius , standing on its circular base.
Water is poured into the tank to a depth of .
 
Calculate the number of litres of water needed.
 

...................................... litres

The diagram shows a hemispherical bowl of radius 5.6 cm and a cylindrical tin of height 10 cm.

Show that the volume of the bowl is 368 cm³, correct to the nearest cm³.
[The volume, , of a sphere with radius is . ]

The tin is completely full of soup.
When all the soup is poured into the empty bowl, 80% of the volume of the bowl is filled.
 
Calculate the radius of the tin.
 

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

The diagram shows a cone with radius 1.75 cm and height 6 cm. 

Calculate the total surface area of the cone.
[The curved surface area, , of a cone with radius and slant height is .] 
 

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

The cone contains salt to a depth of 4.5 cm.
The top layer of the salt forms a circle that is parallel to the base of the cone.

Show that the volume of the salt inside the cone is 18.9 cm³, correct to 1 decimal place.
[The volume, , of a cone with radius and height is . ]  

The salt is removed from the cone at a constant rate of 200 mm³ per second.
 
Calculate the time taken for the cone to be completely emptied.
Give your answer in seconds, correct to the nearest second.
 

............................................. s

The diagram shows a solid prism with length 15.2 cm.
The cross-section of this prism is a regular hexagon with side 7 cm.

Calculate the volume of the prism.
 

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

Calculate the total surface area of the prism.
 

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

A solid metal prism with volume 500 cm³ is melted and made into 6 identical spheres.
 
Calculate the radius of each sphere.
[The volume, , of a sphere with radius  is .]
 

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