VOLUMES OF SOLIDS
By the end of this set of exercises, you should be able to
(a) calculate the volumes of a prism, cone and sphere
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VOLUMES OF SOLIDS
Volumeprism = Areabase x height
A . Volume of a Prism
Exercise 1
1. For each of the following prisms, the area of the base or end face is given.
Calculate the volumes of the prisms:
Area = 29 cm2
Area = 8 cm2
(a)
(b)
Area = 12·5 cm2
(c)
8 cm
10 cm
6 cm
Area = 15·4 cm2
(d)
Area = 52 mm2
(e)
(f)
4·5 cm
11 mm
7 cm
Area = 9·2 cm2
2. This time you must calculate the shaded area first, then find the volumes of the prisms.
10 cm
(a)
7 cm
(b)
6 cm
(c) 7·5 cm
7·5 cm
4 cm
12 cm
5 cm
7 cm
rectangle
(d)
right angled
triangle
square
20 cm
(e)
(f)
15 cm
height =
17 cm
11 cm
6 cm
12 cm
isosceles
triangle
square with
square hole
10 cm
Mathematics Support Materials: Mathematics 1 (Int 2) – Student Materials
circle with
radius = 6 cm
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3. The cylinder – a special prism.
Calculate the volumes of the following cylinders:
(a)
8 cm
(b)
Volume (cylinder) = πr2h
(c)
3 cm
2·5 cm
10 cm
9·5 cm
13 cm
(d)
15 cm
(e)
2 cm
6·5 cm
1 metre
1 cm3 = 1 ml ;
4. Remember:
1000 cm3 = 1000 ml = 1 litre
How many litres of water will the following drums hold?
(a)
25 cm
(b)
40 cm
(c)
60 cm
40 cm
55 cm
35 cm
7 cm
5. A cylindrical tin of Maxcafe Coffee is 10
centimetres high and has a base diameter
of 7 centimetres.
What is the volume of coffee in the tin
when it is full?
10 cm
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6. This rectangular storage tank is full of white paint.
(a) Calculate the volume of paint in the tank
in cubic centimetres (cm3).
(b) Calculate the volume of this cylindrical
paint tin.
50 cm
45 cm
20 cm
80 cm
16 cm
(c) How many times can the paint tin be completely filled from the tank?
7. Meanz Beanz tins are packed into this cardboard box.
(a) How many tins can be placed on the
bottom layer?
8 cm
M
B
33 cm
11 cm
(b) How many layers will there be?
(c) How many tins can be packed in
the box altogether?
(d) How much air space in the box is
there around all the tins?
32 cm
48 cm
8. This cast iron pipe has an internal diameter of 16 centimetres and an outside diameter of 20
centimetres. The pipe is 1·5 metres long.
16 cm
20 cm
1·5 m
Calculate the volume of iron needed to make the pipe.
9. How much liquid feeding will this
semi-cylindrical pig-trough hold?
18 cm
120 cm
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B . Volume of a Cone
Volume (cone) = 1/3πr2h
Exercise 2
1. Calculate the volumes of the following conical shapes:
(a)
(b)
(c)
10 cm
15 cm
6 cm
(d)
18 cm
7 cm
3·5 cm
(e)
16 cm
10·8 cm
40 cm
12·6 cm
6 cm
2. The wafer of an ice-cream cone
has a diameter of 6 centimetres.
The cone is 10 centimetres high.
Calculate the volume of the cone.
10 cm
3.
The ‘sloping’ height of this cone is 26 cm.
The base radius is 10 cm.
26 cm
(a) Calculate the height of the cone.
(b) Calculate the volume of the cone.
10 cm
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4. Calculate the total volumes of the following shapes.
(a)
(b)
30 cm
25 cm
20 cm
30 cm
40 cm
30 cm
18 cm
40 cm
5. Water is poured into this conical flask
at the rate of 50 millilitres per second.
12 cm
(a) Calculate the volume of the flask.
(b) How long will it take, to the nearest
second, to fill the flask to the top?
24 cm
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C . Volume of a Sphere
Volume (sphere) = 4/3πr3
Exercise 3
1. Calculate the volumes of the following spheres:
(a)
(b)
(c)
11 cm
9·2 cm
6·5 cm
(d)
(e)
10·4 cm
30 cm
2. This football is fully inflated.
Calculate the volume of air
inside the football.
24 cm
3. Calculate the volumes of these two ‘hemispheres’:
(a)
(b)
14 cm
8·5 cm
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4. (a) Calculate the volume of water which can be
stored in this copper hot water tank in cm3.
The tank consists of a cylinder with two
hemispherical ends.
40 cm
(b) How many litres of water will it hold?
(1cm3 = 1 ml; 1000 ml = 1 litre).
5.
11 cm
60 cm
Calculate the volume of this child’s
rocking toy which consists of a cone
on top of a hemisphere.
7 cm
6. This decorative wooden fruit bowl
is in the shape of a hollowed out
hemisphere.
18 cm
16 cm
Calculate the volume of wood required
to make it.
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Checkup for Volumes of Solids
1. Calculate the volumes of the following prisms:
(a)
Area = 12·5 cm2
(b)
(c)
Area = 28 cm2
6 cm
9 cm
10 cm
Area = 18·5 cm2
2. Calculate the shaded areas and use them to find the volume of each shape.
(a)
12 cm
(b)
9 cm
8·5 cm
(c)
22 cm
height =
13 cm
8 cm
right angled
triangle
15 cm
8 cm
7 cm
isosceles
triangle
Vol (cylinder) = πr2h
Vol (cone) = 1/3πr2h
Vol (sphere) = 4/3πr3
3. Calculate the volumes of the following shapes:
(a)
9 cm
6 cm
(b)
15 cm
(c)
11 cm
10·4 cm
7 cm
4. This shape consists of a cone, a cylinder and a hemisphere. Calculate its total volume.
12 cm
18 cm
30 cm
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