IGCSEMaths: ExtendedCIEExam Questions3. Geometry3.7 Volume & Surface Area

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First teaching 2021

Last exams 2024

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Volume & Surface Area (CIE IGCSE Maths: Extended)

Exam Questions

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3.7 Volume & Surface Area

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3.7 Volume & Surface Area

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1
Sme Calculator2 marks

cie-igcse-2020-oct-nov-p4-tz3-q6a

ABCDEFGH is a cuboid.
AB = 8 cm, BC = 5 cm and CG = 11 cm.

Work out the volume of the cuboid.
 

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

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2
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i)
Calculate the external curved surface area of a cylinder with radius 8 m and height 19 m.

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

ii)
This surface is painted at a cost of $0.85 per square metre.

Calculate the cost of painting this surface.

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

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3
Sme Calculator3 marks

cie-igcse-2019-oct-nov-p4-tz2-q4a

The diagram shows a hemisphere with radius 6 cm.

Calculate the volume.
Give the units of your answer.
[The volume, V, of a sphere with radius r is V equals 4 over 3 straight pi r cubed.]

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4
Sme Calculator4 marks

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.

i)
Show that 90 million litres of water flows out in 20 minutes.
 
 

[1]

ii)
Beneath the surface, the lake has vertical sides.

Calculate the drop in the water level of the lake when the gates are open for 20 minutes.
Give your answer in centimetres.
[1 hectare = 104 m2, 1000 litres = 1 m3]
 
 

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

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5
Sme Calculator4 marks

cie-igcse-2020-may-jun-p4-tz3-q6b

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

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6
Sme Calculator4 marks

cie-igcse-2018-may-jun-3-p4-q7a

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

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7
Sme Calculator3 marks

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

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8
Sme Calculator1 mark

A cube has side length x cm. The volume of the cube is 1000 cm3.
 
Calculate the value of x.
 

x = ....................................................

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9
Sme Calculator2 marks

cie-igcse-2018-oct-nov-p4-tz2-q10b

A solid metal cone has radius 10 cm and height 36 cm.

Calculate the volume of this cone.
[The volume, V, of a cone with radius r and height h is  V equals 1 third straight pi r squared h.]
 

 

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

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1a
Sme Calculator4 marks

cie-igcse-2020-may-june-p4-tz1-q9aThe diagram shows a sector of a circle with centre O, 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, A, of a sphere with radius r is  A equals 4 pi space r squared. ]


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

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1b
Sme Calculator6 marks

cie-igcse-2020-may-june-p4-tz1-q9c

A cone is made from the sector by joining O A to O B.

i)
Calculate the radius, r, of the cone.

r equals ........................................... cm [2]
ii)
Calculate the volume of the cone.
[The volume, V, of a cone with radius r and height h is  V equals 1 third pi space r squared h. ]

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

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2a
Sme Calculator2 marks

Show that the volume of a metal sphere of radius 15 cm is 14 140 cm3, correct to 4 significant figures.
[The volume, V, of a sphere with radius r is V space equals space 4 over 3 space pi space r cubed.]

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2b
Sme Calculator5 marks
i)
The sphere is placed inside an empty cylindrical tank of radius 25 cm and height 60 cm.
The tank is filled with water.
cie-igcse-2020-specimen-p4-q10b
Calculate the volume of water needed to fill the tank.
 
......................................... cm3 [3]
ii)
The sphere is removed from the tank.
cie-igcse-2020-specimen-p4-q10bii
Calculate the depth, d, of water in the tank.
 
d space equals ......................................... cm [2]

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3
Sme Calculator5 marks

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 h cm.

i)
Show that h equals 13.5.
[The volume, V, of a sphere with radius r is V equals 4 over 3 straight pi r cubed.]
[The volume, V, of a cone with radius r and height h is V equals 1 third straight pi r squared h.]

[2]

ii)
Calculate the slant height of the cone.

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

iii)
Calculate the curved surface area of the cone.
[The curved surface area, A, of a cone with radius r and slant height l is A equals straight pi r l.]

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

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4
Sme Calculator6 marks

cie-igcse-2019-oct-nov-p4-tz2-q4b

The diagram shows a prism ABCDEF.
The cross-section is a right-angled triangle BCD.
BD = 10 cm, BC = 5.2 cm and ED = 18 cm.

Work out the volume of the prism.
 

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

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5a
Sme Calculator3 marks

cie-igcse-2019-may-jun-p4-tz1-q5a

 
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

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5b
Sme Calculator3 marks

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

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5c
Sme Calculator3 marks

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

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6
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cie-igcse-2019-oct-nov-p4-tz3-q6b---edit

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, V, of a sphere with radius r is V equals 4 over 3 straight pi r cubed .]
 

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

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7a
Sme Calculator3 marks

cie-igcse-2018-oct-nov-p4-tz1-q5

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 h cm.

Show that the value of h is 4.7 .

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7b
Sme Calculator6 marks

Calculate the total surface area of the prism.
 

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

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8a
Sme Calculator6 marks

cie-igcse-2018-oct-nov-p4-tz3-q3a

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

i)
Calculate the curved surface area of the cone.
[The curved surface area, A, of a cone with radius r and slant height l is A equals pi r l.]
 
 

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

ii)
Calculate the volume of the cone.
[The volume, V, of a cone with radius r and height h is V equals 1 third straight pi r to the power of italic 2 h.]
 
 

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

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8b
Sme Calculator2 marks
i)
The cone is made of wood and 1 cm3 of the wood has a mass of 0.8 g.
 
Calculate the mass of the cone.
 
 

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

ii)
The cone is placed in a box.
The total mass of the cone and the box is 1.2 kg.

Calculate the mass of the box.
Give your answer in grams.
 
 

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

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9a
Sme Calculator4 marks

cie-igcse-2018-oct-nov-p4-tz3-q3b


The diagram shows a solid cylinder and a solid sphere.
The cylinder has radius 3 r and height 8 r.
The sphere has radius r.

Find the volume of the sphere as a fraction of the volume of the cylinder.
Give your answer in its lowest terms.
[The volume, V, of a sphere with radius r is V equals 4 over 3 straight pi r cubed .]

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9b
Sme Calculator4 marks

The surface area of the sphere is 81straight pi cm2.
 
Find the curved surface area of the cylinder.
Give your answer in terms of straight pi.
[The surface area, A, of a sphere with radius r is  A equals 4 straight pi r squared.]
 

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

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10a
Sme Calculator3 marks

A solid hemisphere has volume 230 cm3.

Calculate the radius of the hemisphere.
[The volume, V, of a sphere with radius r is V equals 4 over 3 pi r cubed.]

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

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10b
Sme Calculator3 marks

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

cie-igcse-2018-may-jun-1-6b

The total volume of the toy is 300 cm3.

Calculate the height of the cylinder.


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

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11a
Sme Calculator3 marks

A sphere has radius x cm. The volume of the sphere is 1000 cm3.
 
Calculate the value of x.
[The volume, V, of a sphere with radius r is V equals 4 over 3 straight pi space r cubed.] 
 

x equals..........................................................

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11b
Sme Calculator4 marks

cie-igcse-2019-may-jun-p4-tz1-q10d

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 x.
 

x equals...................................................

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12
Sme Calculator3 marks

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

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1a
Sme Calculator3 marks

A cylinder with radius 6 cm and height h cm has the same volume as a sphere with radius 4.5 cm.
 
Find the value of h.
[The volume, V, of a sphere with radius r is space V equals 4 over 3 straight pi r cubed.]
 

h equals................................................

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1b
Sme Calculator3 marks

A solid metal cube of side 20 cm is melted down and made into 40 solid spheres, each of radius r cm.
 
Find the value of r.
[The volume, V, of a sphere with radius r is space V equals 4 over 3 straight pi r cubed.]
 

r equals................................................

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2
Sme Calculator3 marks

A solid cylinder has radius x cm and height fraction numerator 7 x over denominator 2 end fraction cm.
The surface area of a sphere with radius R cm is equal to the total surface area of the cylinder.
 
Find an expression for R in terms of x.
[The surface area, A, of a sphere with radius r is space A equals 4 straight pi r squared.]
 

R equals................................................

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3a
Sme Calculator2 marks

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, A, of a cone with radius r and slant height l is A equals pi r l.]


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

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3b
Sme Calculator4 marks

Calculate the volume of the cone.
[The volume, V, of a cone with radius r and height h is V equals 1 third pi r to the power of italic 2 h.]


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

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3c
Sme Calculator4 marks

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, V, of a sphere with radius r is V equals 4 over 3 pi r cubed.]

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4
Sme Calculator5 marks

The diagram below shows a solid circular cone and a solid sphere.

cie-igcse-2020-specimen-p4-q10c

The cone has radius 5 x cm and height 12 x cm.
The sphere has radius r cm.
The cone has the same total surface area as the sphere.

Show that  r squared space space equals space 45 over 2 x squared.

[The curved surface area, A, of a cone with radius r and slant height l is A equals pi r l.]
[The surface area, A, of a sphere with radius r is A equals 4 pi r squared.]

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5
Sme Calculator4 marks

cie-igcse-2019-may-jun-p4-tz1-q10c

A cone has radius x cm and slant height x square root of 5 cm. The volume of the cone is 1000 cm3.
 
Calculate the value of x.
[The volume, V, of a cone with radius r and height h is V equals 1 third straight pi space r squared space h.] 
 

x equals ....................................................

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6a
Sme Calculator3 marks

The volume of a solid metal sphere is 24430 space text cm end text cubed.

Calculate the radius of the sphere.
[The volume, V, of a sphere with radius r is V equals 4 over 3 straight pi space r cubed.] 
 

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

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6b
Sme Calculator3 marks

The metal sphere is placed in an empty tank.
The tank is a cylinder with radius 50 space text cm end text, standing on its circular base.
Water is poured into the tank to a depth of 60 space text cm end text.
 
Calculate the number of litres of water needed.
 

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

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7a
Sme Calculator2 marks

cie-igcse-2019-may-jun-p4-tz3-q4a

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, V, of a sphere with radius r is V equals 4 over 3 straight pi space r cubed. ]

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7b
Sme Calculator4 marks

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

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8a
Sme Calculator5 marks

cie-igcse-2019-may-jun-p4-tz3-q4b

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, A, of a cone with radius r and slant height l is A equals straight pi r l.] 
 

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

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8b
Sme Calculator4 marks

cie-igcse-2019-may-jun-p4-tz3-q4bii

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, V, of a cone with radius r and height h is V equals 1 third straight pi space r to the power of italic 2 h. ]  

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8c
Sme Calculator3 marks

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

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9a
Sme Calculator2 marks

cie-igcse-2018-may-jun-3-p4-q7b


The diagram shows three solids.
The base radius of the cone is 6 cm and the slant height is 12 cm.

Show that the total surface area of the cone is 108straight pi cm2.
[The curved surface area, A, of a cone with radius r and slant height l is A equals straight pi r l.]

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9b
Sme Calculator4 marks

The radius of the sphere is x cm and the radius of the hemisphere is y cm.
The total surface area of each solid is the same.
  
Find the value of x and the value of y.
[The surface area, A, of a sphere with radius r is A equals 4 straight pi r squared.]

x equals..............................................      
y equals...............................................     

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10a
Sme Calculator5 marks

cie-igcse-2018-march-p4-q5a

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

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10b
Sme Calculator3 marks

Calculate the total surface area of the prism.
 

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

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11
Sme Calculator3 marks

A solid metal prism with volume 500 cm3 is melted and made into 6 identical spheres.
 
Calculate the radius of each sphere.
[The volume, V, of a sphere with radius r is V equals 4 over 3 straight pi r cubed.]
 

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

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12
Sme Calculator6 marks

cie-igcse-2020-oct-nov-p4-tz2-q9a

The diagram shows a prism with a rectangular base, ABFE.
The cross-section, ABCD, is a trapezium with AD = BC.
AB = 8 cm, GH = 5 cm, BF = 12 cm and angle ABC = 70°.

Calculate the total surface area of the prism.

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

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