Scale factors and area

We're told that polygon Q is a scaled copy of polygon P using a scale factor of one half. Polygon Q's area is what fraction of polygon P's area? Pause this video and see if you can figure that out.

All right, my brain wants to make this a little bit tangible. Once we get some practice, you might be able to do it without drawing pictures. But they're saying some arbitrary polygon Q and P, so let's just make a simple one. Since we're talking about area, I like to deal with rectangles because it's easy to think about the areas of rectangles.

Let's see. Polygon Q is a scaled copy of polygon P. So let's start with polygon P, and I will do this in red. So polygon P, let's just say I'm going to create an arbitrary polygon. Let's say that this side right over here is 4, and this side right over here is equal to 8. This is polygon P right over here. It's a quadrilateral; it's in fact a rectangle, and its area is just going to be 4 times 8, which is 32.

Now let's create polygon Q. Remember, polygon Q is a scaled copy of P using a scale factor of one half. We're going to scale it by one half. So instead of this side being 4, it's going to be 2. Instead of this side over here, this being 8, the corresponding side in the scaled version is going to be 4. So there you go, we've scaled it by one half.

Now, what is our area going to be? Well, our area, and this is polygon Q, so our area is going to be 2 times 4, which is equal to 8. Notice that polygon Q's area is one-fourth of polygon P's area. That makes sense because when you scale the dimensions of the polygon by one half, the area is going to change by the square of that. One-half squared is one-fourth, so the area has been changed by a factor of one-fourth.

Or another way to answer this question: Polygon Q's area is what fraction of polygon P's area? Well, it's going to be one-fourth of polygon P's area. The big takeaway here is if you scale something—if you scale the sides of a figure by one half each—then the area is going to be the square of that. So one half squared is one over four. If it was scaled by one third, then the area would be scaled or the area would be one ninth. If it was scaled by a factor of 2, then our area would have grown by a factor of 4.

Let's do another example here. We are told rectangle N has an area of 5 square units. Let me do this in a different color. So rectangle N has an area of five square units. James drew a scaled version of rectangle N and labeled it rectangle P, so they have that right over here. This is a scaled version of rectangle N.

What scale factor did James use to go from rectangle N to rectangle P? So let's think about it. We give us rectangle P right over here, and let's think about its dimensions. This height is one, two, three, four, five; it's five high, and it is one, two, three, four, five, six, seven, eight, nine wide. So its area, its area is equal to 45.

Now rectangle N had an area of 5 square units, so our area—let me write this down—N area to P area is multiplying by a factor of nine for going from an area of five square units to 45 square units. Notice N's area is 5. In that color, N's area is 5 square units. P's area we just figured out is 45 square units. So we have it growing by a factor of 9.

Now, what would be the scale factor if our area grew by a factor of 9? Well, we just talked about the idea that area will grow. The factor with which area grows is the square of the scale factor. So, one way to think about it is scale factor squared is going to be equal to nine.

Another way to think about it: our scale factor is going to be equal to 3 to go from N to P. Now let's verify that. We answered their question, but I just wanted us to feel good about it. Let's draw a rectangle that is scaled down from P by a factor of three.

If we were to scale it up by a factor of three, we get rectangle P. So its bottom would have a length of three instead of nine. So it'd be like this: so that would be three, and its height instead of being five, it would be five thirds. Five thirds is one and two thirds, so it'd go about that high. It would look something like that; it would be five thirds.

So our rectangle N would look like this. What is its area? Well, five thirds times three is indeed 5 square units. So notice when we have the area growing by a factor of nine, the scale factor of the size to go from five thirds to five, you multiply by three. To go from three to nine, you multiply by three.