Area and Volumes of Similar Figures
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 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 $k>0$k>0 (this value can actually be less than one, in which case the square would shrink, but let's just look at positive $k$k 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 $a$a units 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 lots of $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.
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.
Question 1
Find the value of $x$x in each of the following cases:
Two rectangles. The larger rectangle, situated on the left, has a length of $27$27 cm with an area labeled as $324$324 $cm^2$cm2. The smaller rectangle, placed on the right, has a length of $9$9 cm and an area denoted as $x$x $cm^2$cm2. The widths of the rectangles are not provided in the image. 
Two hexagons. The larger hexagon on the left has bottom side measuring $21$21 cm, and its area is x $cm^2$cm2. The smaller hexagon on the right has bottom side measuring $7$7 cm, and its area is given as $56$56 $cm^2$cm2. The other dimensions of the hexagons are not specified.
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 a length scale factor of $k>0$k>0.
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 lots of $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.
Question 2
Consider the two similar trapezoidal prisms.
Find the length 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.
Lines and Similarity
As it turns out, when two figures are similar, it's not just the side lengths that are in proportion, it's absolutely any pair of corresponding lines between the shapes. This could include radii, diameters or diagonals. This is true for three dimensional shapes as well.
This is why the length scale factor is often called the linear scale factor. The linear scale factor can be found from the ratio of absolutely any pair of corresponding lines in the figure.
Question 3
The diameter of a circle is tripled. What happens to its area?
It is five times the older area.
AIt is nine times the older area.
BIt is four times the older area.
C
Surface Area Scale Factor
Given what we've just learned about the linear scale factor, we can find the relationship between the linear scale factor and surface area scale factor of a solid.
Similar squares in three dimensional space with a length scale factor $k$k will have an area scale factor $k^2$k2, given what we know about areas of similar figures.
Given that the surface area of a shape is just all the area sum of all the little squares that fit on its surface, whenever we scale a solid using a linear scale factor, all the little square side lengths will scale also.
Hence, the total surface area sum will just scale by $k^2$k2, as was the case in two dimensions!
For any three dimensional figure that is scaled by a linear scale factor of $k$k, the surface area will scale by a surface area scale factor of $k^2$k2.
Question Four
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.