Given the following figure, find the area of a similar polygon 𝐴 prime 𝐵 prime 𝐶 prime 𝐷 prime in which 𝐴 prime 𝐵 prime is equal to six.
So what we’re gonna do here is we’re going to enlarge the shape that we have. But first of all, on our graph, we have 𝐴𝐵𝐶𝐷 which is a rectangle. And we want to find out the area of this. And to find the area of the rectangle, what we do is we multiply the length by the width. And if we take a look at our diagram, we can see that the width is three units. And the length or height is five units. So therefore using the formula we have, we can say that the area of the rectangle 𝐴𝐵𝐶𝐷 is gonna be equal to five multiplied by three which is gonna give a total of 15 units squared.
So now that we have the area of the original shape 𝐴𝐵𝐶𝐷, all we need to look at is the scale factor between the original shape and the new similar enlargement polygon. Now to find the scale factor enlargement, what we do is we divide a new length, so a length on new polygon, by the original length, so the same corresponding length on the original polygon, on the original rectangle. So, therefore, the two corresponding lengths we have are 𝐴 prime 𝐵 prime is equal to six on the new polygon and 𝐴𝐵 is equal to three on the original polygon. So, therefore, the scale factor is gonna be equal to six over three, which is gonna be equal to two.
So we’ve got a scale factor enlargement of two from the original rectangle to the new rectangle. So I’ve drawn a little sketch which shows the new polygon. It’s not to scale. It’s just to give us an idea of what’s happening. So we’ve got the new length 𝐴 prime 𝐵 prime is six. We know that. But to find out the area, we also need to know what 𝐵 prime 𝐶 prime or 𝐴 prime 𝐷 prime is going to be. Well, 𝐵 prime 𝐶 prime and 𝐴 prime 𝐷 prime are gonna be equal to five multiplied by two. And that’s because it’s the length of the original, which is five where 𝐵𝐶 is, and then multiplying it by the scale factor of enlargement.
So this is gonna give us a length of 10 units. Okay, great, so now we have the width and the length of the new polygon. So, therefore, the area of new polygon is gonna be equal to 10 multiplied by six, cause that’s the length multiplied by the width, which is gonna give a final answer of 60 units squared. So we’ve solved the problem and found the area of the similar polygon 𝐴 prime 𝐵 prime 𝐶 prime 𝐷 prime.
There is, however, one other way that we could have solved the problem when we got to the stage where we’d found out the area of the original shape and the scale factor. I’m gonna use that to check the answer. And I’ll show you how it works. So for this check, we’re gonna have a look at the scale factors. So if we’ve got a scale factor for a length, so in this case, it’s two. Then the scale factor of the areas of two shapes is going to be the scale factor of the lengths squared.
And again, if we want to find the scale factor for the volumes, it’s going to be the scale factor cubed. And if we think why that might be? Why we would have a length multiplied by a length. So if there’d been a scale factor involving one of the lengths and then we multiply it by itself and when we look at the area, then it’ll be the scale factor squared. And the same for when we’re looking at the volume in cubed.
So, therefore, we can say that the scale factor of the areas is gonna be equal to two squared which is gonna be equal to four. So, therefore, the area of the new shape is gonna be equal to four multiplied by 15 because that’s the scale factor of the areas multiplied by the area of the original shape which gives us 60 units squared as an answer. So it’s the same as before. And we’ve checked it. And it’s the correct answer to the problem.