14. SURFACE AREA
14 - 1 Surface area in science and technology
2
14 - 2 Calculating surface area
3
14 - 3 Using nets
4
14 - 4 Drawing nets
6
14 - 5 Changing the volume
7
14 - 6 Changing the surface area
8
14 - 7 Surface area, volume and size
10
14 - 8 Designing a chocolate box
13
© 2004, McMaster & Mitchelmore
1
www.workingmaths.net
Activity 14 – 1
Surface area in science and technology
Rabbits put their ears down and have a compact shape on a cold day to
reduce the amount of heat they could lose from their bodies.
A car’s radiator has a large surface area so heat can be quickly lost from
it into the atmosphere.
6 large pizzas keep warmer if their boxes are stacked in the one pile
because this reduces the surface area from which their heat can escape.
© 2004, McMaster & Mitchelmore
2
www.workingmaths.net
Activity 14 – 2
Calculating surface area
A cuboid has 6 faces.
The area of the side at the front:
Area = 2cm x 10cm = 20 cm2
What is the area of the side on top?
Area = 3cm x 10cm = 30 cm2
What is the area of the side at one end?
Area = 2cm x 3cm = 6 cm2
In the space below, calculate the surface area (SA) of the cuboid.
SA = 2 x (20 + 30 + 6) cm2
= 2 x 56 cm2
= 112 cm2
You only need to make 3 different measurements to be able to calculate
the surface area of a cuboid.
The area of paper needed to wrap a present is greater than its surface
area because extra paper is needed for the places where the paper
overlaps.
© 2004, McMaster & Mitchelmore
3
www.workingmaths.net
Activity 14 – 3
Using nets
rectangular prism (cuboid)
triangular prism
pentagonal prism
triangular prism
cube
You tell whether a net is the net of a prism because if it is, all its faces or
all except two of its faces are parallelograms. The remaining two faces
are congruent and they have the same number of sides as the number
of parallel faces.
© 2004, McMaster & Mitchelmore
4
www.workingmaths.net
15mm
30mm
45mm
60mm
15mm
The easiest methods are:
Method 1
Area = (60mm x 90mm) + 2 (15mm x 30mm)
= 5 400 + (2 x 450) mm2
= 5 400 + 900 mm2
= 6 300 mm2
= 63 cm2
Method 2
Area = (90mm x 90mm) - 2 (15mm x 15mm) - 2 (15mm x 45mm)
= (90 x 90) – (2 x 15 x 60) mm2
= 8 100 – 1 800 mm2
= 6 300 mm2
= 63 cm2
© 2004, McMaster & Mitchelmore
5
www.workingmaths.net
Activity 14 – 4
Drawing nets
Method 1
SA = SA base + SA end faces + SA side faces
= 8 x 12 + 2(4 x 8) + 2(4 x 12) cm2
= 96 + 64 + 96 cm2
= 256 cm2
Method 2
The net could be drawn as one large rectangle with a square cut out of
each corner.
Area of the large rectangle: 20 x 16 = 320 cm2
Area of each square: 4 x 4 = 16 cm2
The surface area of the box using these areas:
SA = 320 - (4 x 16 ) cm2
= 256 cm2
© 2004, McMaster & Mitchelmore
6
www.workingmaths.net
Activity 14 – 5
Changing the volume
The surface area of each box:
A: 60 – 4 = 56 B: 60 – 4 = 56 C: 60 – 4 = 56 D: 60 – 4 = 56
The 3 dimensions (ie. width, depth and height) of each box:
A: 4 x 8 x 1
B: 3 x 10 x 1
C: 2 x 13 x 1 D:18 x 1 x 1
The volume of each box:
A: 32
B: 30
C: 26
D: 18
Yes. Two boxes have the same surface area but a different volume.
The shapes cut out of the corners of the cardboard have to be squares
because 2 adjacent sides have to be equal (to form an edge of the box)
and at right angles (so the sides of the box are perpendicular to its
base).
Side of Area of 4
square squares
(cm)
(cm)
SA
of box
(cm2)
Length
of box
(cm)
Width
of box
(cm)
Height
of box
(cm)
Volume
of box
(cm3)
1
4
196
18
8
1
144
2
16
184
16
6
2
192
3
36
164
14
4
3
168
4
64
136
12
2
4
96
A side length of 2 for the cut-out squares gives the tray with the greatest
volume.
No. This box does not also have the greatest surface area.
Yes. You could make a tray with an even greater volume.
You would make the square with a side length a bit more than 2 and
less than 3 eg. 2.1.
Volume = 2.1 x 15.8 x 5.8 = 192.444 cm3
© 2004, McMaster & Mitchelmore
7
www.workingmaths.net
Activity 14 – 6
Changing the surface area
All the factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
Possible cuboid dimensions: 24 x 1 x 1; 12 x 2 x 1; 6 x 4 x 1; 6 x 2 x 2;
8 x 3 x 1; 4 x 2 x 3.
You would expect each cuboid to have the same volume because their
dimensions multiplied together make 24 (their sides being factors of 24).
© 2004, McMaster & Mitchelmore
8
www.workingmaths.net
As the surface area gets smaller, the cuboid gets more compact (or
squarish).
As the surface area gets smaller, the 3 dimensions get closer to each
other in length.
With 12 cubes, the dimensions of the cuboid with the smallest surface
area would be 2 x 2 x 3.
2
2
3
Surface area of the cube above = (2 x 2 x 2) + (4 x 3 x 2)
= 8 + 24
= 32 units2
© 2004, McMaster & Mitchelmore
9
www.workingmaths.net
Activity 14 – 7
Surface area, volume and size
Length of Area of
one edge one face
(units)
(units2)
Surface area
(SA)
(units2)
Volume (V)
(units3)
Ratio
SA
V
6
=6
1
1
1
6x1=6
1x1x1=1
2
4
6 x 4 = 24
2x2x2=8
3
9
6 x 9 = 54
3 x 3 x 3 = 27
4
16
6 x 16 = 96
4 x 4 x 4 = 64
5
25
6 x 25 = 150
5 x 5 x 5 = 125
24
=3
8
54
=2
27
96
= 1.5
64
150
= 1.2
125
As the length of the edges increases, both SA and V increase with V
increasing at a faster rate than SA.
SA
As the cube increases in size, surface area to volume ratio ( V )
decreases.
SA
Toblerone size
Surface area (SA)
(cm2)
Volume (V)
(cm3)
50g
167
69
2.4
100g
236
138
1.7
200g
334
276
1.2
400g
473
552
0.9
Ratio (
V
)
The surface area to volume ratio decreases as toblerone chocolates
increase in size.
© 2004, McMaster & Mitchelmore
10
www.workingmaths.net
150
130 140
70
80
90
100
110
120
•
Surface Area (units 2)
Volume (units3)
0
10 20
30
40
50
60
•
•
•
•
0
1
2
Length of one edge (units)
© 2004, McMaster & Mitchelmore
3
11
4
5
6
www.workingmaths.net
The amount of oak flavour that comes into the wine depends on the
surface area of the barrel compared with the total amount of wine that is
in the barrel. Juice put in smaller barrels produces wine with a stronger
oak flavour because the smaller barrels have a larger surface area to
volume ratio.
The pile of sugar grains will dissolve faster because for their weight, they
have a larger total surface area in contact with the water molecules. The
water molecules therefore interact with more molecules of sugar at a
time and this causes the sugar to dissolve faster.
The blue-tongue lizard would be expected to get moving first in the
morning because it receives more heat through its skin compared with
the volume of the body it has to move.
In a hot place (eg. inside a car with closed windows) a baby’s body
becomes heat-stressed sooner than an adult’s body because, for the
size of its body, it has a greater surface area which absorbs the heat.
During a race, a runner’s body produces a lot of heat. This heat is lost
through perspiration. Marathon runners are usually smaller than other
athletes because smaller people are able to lose more heat through their
skin in relation to their body size.
© 2004, McMaster & Mitchelmore
12
www.workingmaths.net
Activity 14 – 8
Designing a chocolate box
The design specifications are:
Shape of base: a right isosceles triangle
Volume: 69 cm3
Surface area: less than 167 cm2
© 2004, McMaster & Mitchelmore
13
www.workingmaths.net