8.07 Perimeters and areas of similar figures
Recall that two figures are similar if corresponding angles are congruent and corresponding sides are proportional. 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 linear 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, shown in red below, and we enlarge it by a length scale factor of $3$3, shown in blue below.
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.
Exploration
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 less than one, in which case the square would shrink, but let's just look at $k>1$k>1 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.
For any figure that is scaled by a linear scale factor of $k$k, the area will scale by an area scale factor of $k^2$k2.
Practice questions
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.
Question 2
Consider the two similar rectangles shown in the figure.
Find the area of Rectangle A.
Area A = $\editable{}$cm2
Find the area of Rectangle B.
Area B = $\editable{}$cm2
What is the ratio of the area of Rectangle A to Rectangle B?
Is it true in this case that if the matching sides of two similar figures are in the ratio $m:n$m:n, then their areas are in the ratio $m^2:n^2$m2:n2?
True
AFalse
B