Two corresponding sides of two similar polygons have lengths of 54 and 57 centimeters. Given that the area of the smaller polygon is 324 centimeters squared, determine the area of the bigger polygon.
We’re told that these two polygons are similar. Now we know that two polygons are similar, if two things are true. Firstly, all of their corresponding angles must be equal. Secondly, corresponding sides on the two polygons must be in the same ratio. So, they don’t have to be in the same length but they must be in the same ratio. Meaning that, if one side on one polygon is twice as big as it is on the other polygon, all of the other sides must also be twice as big.
This ratio is also called the scale factor, or the length scale factor. And as we’ve been given the lengths of corresponding sides, 54 and 57 centimeters, we can work it out. The length scale factor then, abbreviated to LSF, can be found by dividing a length on the larger polygon by the corresponding length on the smaller polygon. So, that’s 57 over 54.
This fraction can actually be simplified slightly, as both the numerator and denominator are multiples of three. Dividing 57 by three gives 19. And dividing 54 by three gives 18. So, the length scale factor simplifies to 19 over 18.
Now we’re told that the area of the smaller polygon is 324 centimeters squared and asked to work out the area of the bigger polygon. Does this mean that the area of the bigger polygon is just 324 multiplied by the scale factor of 19 over 18? Well, actually no. And the reason for this is that the scale factor for lengths and the scale factor for areas are not the same. Let’s see why.
Suppose we have two squares. Now squares are always similar to one another. Suppose the first square has a side length of one centimeter. And the second has a side length of three centimeters. The length scale factor between these two squares then would be three over one, which is just equal to three.
Let’s also consider the areas of these squares. To find the area of a square, we square its side length. So, for the first square, one centimeter multiplied by one centimeter gives one centimeter squared. For the second square, the area is three centimeters multiplied by three centimeters, which is nine centimeters squared. The scale factor between the areas then, which we can abbreviate to ASF, is nine over one, which is just equal to nine. Now this is not equal to three, the length scale factor, but there is a relationship between these two scale factors. Can you spot what it is?
Well, nine is equal to three squared. And in fact, this relationship is always true. If the length scale factor between two similar polygons is some number 𝑘, then the area scale factor will be that number squared, 𝑘 squared. Which means to find the area of the bigger polygon, we multiply the area of the smaller polygon not by 19 over 18 but by 19 over 18 squared.
Now we can work this out. To square a fraction, we square both its numerator and denominator. 19 squared is 361. And 18 squared is 324. Now spot that we have 324 in the numerator and 324 in the denominator. So, these cancel, leaving just one multiplied by 361 over one. So, the calculation simplifies to 361. The units for this area will be the same as the units for the area of the smaller polygon. That’s centimeters squared.
So, by recalling that if the length scale factor between two similar polygons is 𝑘, then their area scale factor will be 𝑘 squared, we’ve found that the area of the bigger polygon is 361 centimeters squared.