Recall that two figures are similar if they both have the same shape, even if one has been enlarged or shrunk. All pairs of corresponding sides in similar figures are in the same ratio, and we can use this ratio to find the scale factor.
Since this scale factor affects the length of sides, it is also called the length scale factor or linear scale factor.
But what happens to the area of a figure when we enlarge it by a length scale factor? Does it also enlarge? Is it by the same or some other related scale factor?
Suppose we have a square with side lengths $2$2 cm and we enlarge it by a length scale factor of $3$3.
Our lengths have been scaled by a factor $3$3, but our area has gone from $4$4 cm2 to $36$36 cm2. It has been scaled by a factor of $9$9.
Areas of similar figures do not scale by the same factor as the linear scale factor, and for this reason, they have their own scale factor called the area scale factor.
So, is there any way to predict what the area scale factor of a shape will be if we know its length scale factor? The answer is yes, and we'll see why.
Area and surface area scale factor
Area scale factor of a square
Let's say we have a square of side length $a$a units. This square would therefore have area $a^2$a2 units2.
Let's say we want to scale it by a length scale factor of $k>1$k>1 (this value can actually be between zero and one, in which case the square would shrink, but let's just look at an enlargement factor for now).
What is the area now?
The side length of our new square is $a\times k$a×k units. We can use this to figure out the new area.
| New Area | $=$= | $\left(a\times k\right)\times\left(a\times k\right)$(a×k)×(a×k) | units2 |
| $=$= | $\left(a\times k\right)^2$(a×k)2 | units2 | |
| $=$= | $a^2\times k^2$a2×k2 | units2 |
So, if our old area was $a^2$a2 units2 and our new area is $a^2\times k^2$a2×k2 units2, we have scaled our area by a factor of $k^2$k2. Since this works for any side length and any scale factor $k$k, we know that if we scale any square by length scale factor $k$k, the area scale factor will always be $k^2$k2.
Area scale factor of any figure
Remember back to when you were first learning about areas. Area is defined as the amount of unit squares that fit into a two dimensional shape.
So, if we have a shape whose area is $18.79$18.79 cm2, what we really mean is that it can fit $18.79$18.79 $1$1 cm2 squares inside it.
If we enlarge or shrink any figure, no matter how irregular, by a length scale factor of $k$k, the areas of all these squares inside it will each scale by a factor of $k^2$k2.
Hence, the total area sum will also scale by a factor of $k^2$k2. This means that for absolutely any figure that is scaled by a length scale factor of $k$k, the area will always scale by an area scale factor of $k^2$k2.
By extension, since the surface area of a prism or pyramid is composed of various two dimensional figures, then the scaling of of all sides of the prism or pyramid by a scale factor of $k$k also results in the overall surface area being scaled by an area scale factor of $k^2$k2.
For any figure that is scaled by a length scale factor of $k$k, the area will scale by an area scale factor of $k^2$k2.
Practice questions
Question 1
Question 2
The bases of two similar rectangular prisms are: $30$30 cm by $45$45 cm; and $6$6 cm by $9$9 cm.
Find the length scale factor.
Find the surface area scale factor.
Find the surface area of the smaller prism, knowing that the surface area of the larger prism is $4425$4425cm2.
Volume scale factor
Given what you've learnt above, can you guess how the volume scale factor of a three dimensional figure might change depending on the length scale factor $k$k? If your answer is that it will be $k^3$k3, you're correct!
Volume scale factor of a cube
Given a cube of side length $a$a units, its volume would be $a^3$a3 units3.
Again, let's say we want to scale it by enlarging using a length scale factor of $k$k.
The side length of our new cube is $a\times k$a×k units. We can again use this to figure out the new volume.
| New Area | $=$= | $\left(a\times k\right)\times\left(a\times k\right)\times\left(a\times k\right)$(a×k)×(a×k)×(a×k) | units3 |
| $=$= | $\left(a\times k\right)^3$(a×k)3 | units3 | |
| $=$= | $a^3\times k^3$a3×k3 | units3 |
Hence, we know that if we scale any cube by length scale factor $k$k, the volume scale factor will always be $k^3$k3.
Volume scale factor of any figure
The same principal applies here as before. Volumes are just measured in unit cubes. If we have a three dimensional figure whose volume is $253.79$253.79 cm3, what we really mean is that it can fit $253.79$253.79 $1$1 cm3 cubes inside it.
If we enlarge or shrink any figure, no matter how irregular, by a length scale factor of $k$k, the volumes of all these cubes inside it will each scale by a factor of $k^3$k3.
Hence, the total volume sum will also scale by a factor of $k^3$k3. Again, this means that for absolutely any figure that is scaled by a length scale factor of $k$k, the volume will always scale by an volume scale factor of $k^3$k3.
For any figure that is scaled by a length scale factor of $k$k, the volume will scale by a volume scale factor of $k^3$k3.
Practice questions
Question 3
Consider these two rectangular prisms:
Are the two rectangular prisms similar?
Yes
ANo
BFind the length scale factor going from the smaller prism to the larger prism.
Find the surface area scale factor going from the smaller prism to the larger prism.
Find the volume scale factor going from the smaller prism to the larger prism.
Question 4
The following 2 solids are similar. If the volume of the smaller solid is $193.7$193.7cm3 and the volume of the second is $24212.5$24212.5cm3, find the value of $x$x.
Question 5
The ratio of the length of a model car to a real car is $1:20$1:20.
Find the ratio of surface area of the model car to the real car.
Find the ratio of the volume of the model car to the real car.
Find how many litres of paint are needed to paint the real car, if $18$18 mL are needed to paint the model car.
Find the capacity of the model car fuel tank, in mL, if the real car fuel tank holds $48$48 L.