GRADE 10
MEASUREMENT
VOLUME AND SURFACE AREA
You will already have used at the following formulae:
Volume = area of base  perpendicular height
Surface Area = 2  area of base + perimeter of base  perpendicular height
The set of formulae in the table can be used for any right prism, the most common ones you will use will
be the following:
Shape
Rectangular prism
Cylinder
Volume
V  lbh
V  r 2 h
Surface Area
SA  2lb  2l  2bh
SA  2r 2  2rh
Task 1
1.
Find the volume and surface area of each of the shapes below.
a)
b)
12 cm
15 cm
9 cm
8 cm
6 cm
c)
d)
19 mm
25 mm
5 mm
12 mm
2.
16 mm
A small fish tank has a length and a breadth both of which are 30 cm and a height of 25 cm. The
water level goes up to 23 cm. If the tank is turned on its side, how far up does the water reach?
3.
A tin of baked beans has a diameter of 7 cm. The volume is 450 cm3. Find the height of the tin.
4.
If you need 250 grams of dough to make a 30 cm diameter pizza, how much dough will you need
to make a 40 cm pizza with a crust of the same thickness?
5.
a)
Another fish tank has a regular hexagonal base with each side equal to 8 cm. The height is
25 cm. Find the volume of water in the tank.
b)
The dimensions given in a) are the outside dimensions. If the glass is 2 mm thick, find the
volume of glass used in the tank.
6.
A chocolate is made in a box that has an equilateral triangle as its base. Find the volume of
chocolate in the box.
h
21 cm
3,5 cm
7.
The dimensions of a swimming pool are as shown in the picture.
15 m
2,5 m
1,2 m
5m
4m
6m
8.
9.
a)
Calculate the volume of the pool.
b)
The water level is 10 cm from the top of the pool. If 3 kilolitres of water is lost through
evaporation, how far from the top of the pool is the water surface now?
A cylinder of cheese with radius 15 cm and height 10 cm has a wedge of 120° cut out of it.
a)
Calculate the volume of cheese that has been cut out.
b)
Calculate the surface area of cheese remaining.
The cross section of a loaf of bread consists
of a rectangle and a semi circle.
Calculate the volume of bread.
8cm
30cm
10cm
10.
A cylinder is constructed according to the following specification: the sum of the radius and the
height is 10 cm. Investigate what the dimensions must be to give maximum volume.
THE RATIO OF LENGTHS, AREAS AND VOLUMES
Task 2
1.
Let us work with two cubes; one of side 1 cm and the other of side 2 cm.
a)
Find the volume of each.
b)
Find the surface area of each.
c)
Compare you answers.
2.
What happens if the cubes have sides 1 cm and 3 cm? Do the same as above for these.
3.
Compare the volumes and surface areas for the following pairs of cylinders.
4.
Cylinder A
Cylinder B
a)
r = 1 cm, h = 1 cm
r = 1 cm, h = 2 cm
b)
r = 1 cm, h = 1 cm
r = 2 cm, h = 1 cm
c)
r = 1 cm, h = 1 cm
r = 2 cm, h = 2 cm
d)
r = 1 cm, h = 1 cm
r = 5 cm, h = 8 cm
Look at all your answers and write down what happens to the volume and surface area when you
a)
change the measurement of one dimension
b)
change the measurement of two dimensions
c)
change the measurement of three dimensions
If you have similar figures, the dimensions of their sides, their areas and their volumes are in proportion.
If the lengths of the sides increase x times, then the area increases x 2 times.
If the lengths of the sides increase x times, then the volume increases x 3 times.
Worked Example
Two similar bottles have heights 25 cm and 15 cm. What is the ratio of their volumes?
Lengths (i.e. one dimension)
25:15
3
3
Volume (i.e. three dimensions)
25 :15 = 15625:3375
1.
A circle of radius 8 cm has an area of 201,06 cm2. What is the area of a circle with radius 4 cm?
2.
Two triangles are similar. The base of the smaller triangle is 6 cm and its area is 30 cm2. The
larger triangle has a base of 18 cm. Find its area.
3.
Two rhombuses are similar. The side of the larger is 12 cm and its area is 86 cm2. The area of the
smaller rhombus is 53 cm2. What is the length of one side?
4.
Two similar shapes have heights 4 cm and 8 cm.
(a)
The volume of the smaller shape is 250 cm3. Find the volume of the bigger shape.
(b)
The surface area of the bigger shape is 100 cm2. Find the surface area of the smaller shape.
5.
A cylinder has a height of 16 cm and a volume of 368 cm3. A similar cylinder has a volume of 157
cm3. Find the height of that cylinder.
6.
The capacity of an oval ball is 1024 cm3. A smaller similar ball has a capacity of 640 cm3. The
height of the larger ball is 16 cm. Calculate the height of the smaller ball.
7.
A gold statue has a height of 1 m. It is melted down and similar statues are recast with the gold
with heights of 50 cm. How many statues can be made from the original one?
8.
a)
b)
Determine the volume of water that the mug can
hold, if it has a depth of 9 cm and an internal
diameter of 8 cm.
8 cm
Sinazo would like to double the volume of the water
in the mug. Should she:
i)
ii)
iii)
9 cm
double the radius?
double the height?
double the diameter?
Choose the correct option and the show that your selection is correct.
9
10.
a)
Calculate the volume of the Red bull tin if it has a radius of
25 mm and a height of 128 mm.
b)
The tin holds 250 ml of Red Bull. What is the volume of air in the
tin? (1 ml = 1 cm³)
c)
If the manufacturers decide to make a tin which has a diameter of
100 mm, how tall must the tin be to keep the same volume as in a)?
A cylinder has volume 1440 cubic centimetres. Its height is twice its radius. Determine the
height of the cylinder in centimetres.
11.
12.
4
The volume of a sphere is given by the formula V   r 3 .
3
a)
Change the subject of the given formula to r.
b)
Determine the radius in centimetres of a spherical balloon which contains 65 450 ml of
helium gas. (Recall: 1cm 3 = 1 ml)
c)
A cylinder has volume 1440 cubic centimetres. Its height is twice its radius. Determine
the height of the cylinder in centimetres.
The formula for surface area of a cylinder is:
A  2r 2  2rh
Make the height (h) the subject of the formula.
13.
A factory makes jam tins with dimensions as shown in
the diagram.
a)
Calculate the volume of jam in 1 tin.
b)
How many tins can be packed in a cardboard box
with a length of 28cm, a breadth of 14cm and a
height of 16 cm?
7cm
8cm
16cm
28cm
c)
14cm
By how much will the volume of the cardboard box increase if the length and the breadth
of the box is doubled and the height remains the same?
14.
A rectangle has a length of 16m and a breadth of 8m. If the length is doubled and the breadth is
halved, by how much has the area changed?
15.
The cylinder alongside has height h and radius r.
a)
Find the volume of the cylinder if r  3cm and
h  10cm .
b)
What happens to the volume of the cylinder if the
radius is multiplied by a factor of 3?
c)
What happens to the volume of the cylinder if both the
height and the radius are doubled?
r
h
16.
A can of Coke contains 340 ml (1 ml = 1
cm3) of liquid.
a)
Assuming the can is a perfect
cylinder of diameter 6.5 cm,
calculate the height of a can of Coke.
b)
What will the new volume of the can
be if the height is doubled and the
diameter is halved?
c)
What will be the ratio of the volume
of the new can to the volume of the
old can?
Height,
h cm
6.5 cm
17.
A wooden block with a square base of x units and a
length of y units has a hole with radius r drilled through
it.
a)
b)
18.
y
Find a formula for the volume (V) of the wood left
once the hole has been drilled through the block.
If the length of the sides of the square and the
radius of the circle are doubled, by what factor
will the volume increase?
x
x
r
A rectangle has a perimeter of 16 cm. If we let the breadth of the rectangle be x, find the length of
the rectangle (y) for each of the given x values. Then find the area of the rectangle formed with the
given dimensions. Use the table to help you. (4)
x (breadth)
1
2
3
y (length)
Area of
rectangle
a)
When does the rectangle have the greatest area?
b)
What do you notice about the rectangle with the greatest area?
c)
Do you think this is always true? Why or why not?
4
19.
Picture A shows a household tin of pineapple slices.
Its height is 12 cm and the radius of its circular base is
6 cm. Picture B shows a catering-size tin of pineapple
slices, with height 48 cm and base radius 20 cm.
48
cm
6 cm
a)
Show that the two cans are not similar in
shape.
b)
If the weight of the contents of can A is 450 g,
calculate the weight of the contents of can B.
12
cm
20 cm
20.
22.
B
c)
If can A is doubled in height but its radius stays the same, what will be the weight of the
contents of the new can?
d)
If can A has its height doubled and its base radius trebled, what will be the weight of the
contents of the new can?
The following diagram shows
a right regular pyramid with a
square base. AMˆ N   , the
angle which the triangular
faces make with the base. H
is the height of the pyramid,
and h is the slant height of the
triangular faces.
A
H
If h  20 cm and
BC  14 cm , calculate each of
the following to TWO decimal
places:
21.
A
a)

b)
H
c)
The total surface area
of the pyramid
d)
The volume of the pyramid
h
D

M
N
B
C
4
The volume of a sphere is given by the formula: V   r 3 . If the radius of a sphere is increased
3
by a factor of 2, determine by what factor the volume will increase.
A round tin has a radius of 15 cm and a height of
8 cm.
a)
Determine the volume of the tin.
b)
If the radius of another round tin was 10
cm, what would the height of the tin need
to be to have the same volume?
23.
A company manufactures solid steel cylinders. Each
cylinder has a radius of 15 cm and a height of 40 cm.
Each cylinder needs to be coated with a special
protective varnish that costs 50c per 500 cm2.
a)
If the radius and height are each doubled,
complete the following table:
Formula
Original cylinder
Radius
r
15 cm
Height
h
40 cm
Area of top/bottom
surface
r2
706,9 cm2
Area of curved surface
b)
“Doubled” cylinder
3769,9 cm2
Hence determine how much extra must be spent coating each “doubled” cylinder.
ANSWERS
Task 1
1a)
V = 720 cm3; SA = 516 cm2
b)
V = 3053,63 cm3; SA = 1187,52 cm2
3
2
c)
V = 570 mm ; SA = 630 mm
d)
V = 2513,27 mm3; SA = 1229,38 mm2
2.
27,6 cm
3.
11,69 cm
4.
444,44 g
5a)
4156,92 cm3
b)
1247,31 cm2
6.
111,39 cm3
3
7a)
111,2 m
b)
15 cm
8a)
2356,19 cm3
b)
189438,04 cm2
9.
3578,10 cm3
10.
20/3 cm
Task 2
1a)
1 cm3; 8 cm3
b)
6 cm2; 24 cm2
c)
volume is 8 times and surface area is 4 times
2a)
1 cm3; 27 cm3
b)
6 cm2; 54 cm3
c)
V is 27 times and SA is 9 times
3a)
3,14 cm3; 12,57 cm2; 6,28 cm3; 18,85 cm2
b)
12,57 cm3; 37,70 cm2
c)
25,13 cm3; 50,27 cm2
d)
628,32 cm3; 408,41 cm2
Task 3
1.
50,27 cm2
2.
90 cm2
3.
9,42 cm
3
4a)
2000 cm
b)
25 cm2
5.
12,04 cm
6.
13,68 cm
7.
8