Scale factors for area and volume
Here are two similar cuboids.
The scale factor is 2.
Does this mean that the larger cuboid has double the
volume of the smaller one?
NO, because the volume actually gets doubled three
times, once for each dimension:
×2 width
×2 depth
×2 height
2×2×2 = 23 = 8. So the larger cuboid actually has 8 times the volume of the smaller
one.
(If the scale factor had been, for instance, four instead of two, the volume would
have been quadrupled for every dimension.
So the volume scale factor would have been 4×4×4 = 43 = 64)
REMEMBER: volumes always scale by the cube of the length scale factor.
Scale factors for area
Look at the nearest face of each cuboid: you will see that it measures 3 by 5 in the
original, and 6 by 10 in the scaled up version.
It has doubled in width and in height, so the area has increased by a factor of 22 = 4.
REMEMBER: areas always scale by the square of the length scale factor.
1. Here are two cuboids with their measurements labelled.
3h
h
w
d
3w
3d
(a) Write down an expression for the volume of the small cuboid.
(b) Write down a simplified expression for the volume of the large cuboid.
(c) Write down an expression for the surface area of the small cuboid.
(d) Write down an expression for the surface area of the large cuboid.
2. (a) Cuboid A has width a, length b and height c.
Write down an expression for its volume.
(b) Cuboid B is similar to A. The scale factor is 5.
Write down an expression for the volume of B.
Here, once again, is what you need to know:
 The area scale factor is the square of the length scale factor.
(This is true whether your shapes are 2D or 3D.)
 The volume scale factor is the cube of the length scale factor.
3. The table has some example scale factors. Fill in the gaps.
Length scale factor
2
3
4
Area scale factor
4
9
Volume scale factor
8
27
36
1000
2500
4. Here are two similar shapes.
(a) Find the volume and the surface
area of the smaller shape.
(b) By using the rules for area and volume scale factors, find the volume and the
surface area of the larger shape.
5. From The BFG, by Roald Dahl:
‘A man does not rise to become the Queen’s butler unless he is
gifted with extraordinary ingenuity, adaptability, versatility,
dexterity, cunning, sophistication, sagacity, discretion and a host
of other talents that neither you or I possess. Mr Tibbs had them
all…In a split second he had made the following calculations in
his head: if a normal six foot high man requires a three-foot high
table to eat off, a twenty-four foot giant will require a twelvefoot-high table.
And if a six-foot man requires a chair with a two-foot-high seat, a twentyfour-foot giant will require a chair with an eight-foot-high seat.
Everything, Mr Tibbs told himself, must be multiplied by four. Two
breakfast eggs must become eight. Four rashers of bacon must become sixteen.
Three pieces of toast must become twelve, and so on. These calculations about
food were immediately passed on to Monsieur Papillion, the royal chef.’
What do you think of the butler’s calculations?
6. This model trebuchet (a medieval catapult for
siege warfare) is one twentieth the size of the real thing.
(a) The model is 56cm long.
How long was the real trebuchet, in metres?
(b) If the toy throws projectiles weighing 17 grams,
suggest how heavy the projectiles of the real trebuchet
would have been.
(c) Summarise the emotions you would feel if under
bombardment by the full size trebuchet.
Hint: Assuming that two similar
objects are made of the same
material, the mass multiplies by
the same as the volume.
7. Two common types of cannonball on old warships were the
4 pounder and 32 pounder, referring to the weight of the
cannonball.
How many times bigger was the diameter of the larger
cannonball (and therefore the calibre of the cannon)?