Scaling perimeter and area example 1 | Transformational geometry | Grade 8 (TX) | Khan Academy

We're told Pentagon A was dilated by a scale factor of three to create Pentagon B. Complete the missing measurements in the table below, so pause this video, have a go at this before we do this together.

All right, now let's work on this together. It's really just trying to ask us how does perimeter or area scale when you have a scaling factor. Because if we know that, we can figure out what these squares are. To help us with that, I will do a simpler figure first before we think about a pentagon, although the same principles will apply.

Let's imagine we have a rectangle. So let's say this is our—actually, let me make it a little bit smaller. Let's say this is our original rectangle right over here, and let's say the long side, the length there, is two, and the short side, the height there, is one. So the perimeter, I'll just write P for perimeter right over here, it's going to be 2 + 2, which is 4, + 1 + 1, it is 6.

Now, what is its area? Its area is going to be 2 * 1, it's going to be 2 square units. Now let's imagine that we dilate it by a scale factor of three. So then what are we going to be looking at? Well, then we would be looking at something that looks like this. Let me do this in a different color, and I'm not obviously going to be able to do it exact, but it would look something like this.

So that side was one; now that side is going to be three. We've scaled it up by three. This side was two; now this side is going to be six. Similarly, that side was one; now that side's going to be three. Let me just complete the rectangle there. So now when we scale it up, when we have a scale factor of three, what is our new perimeter?

Now our new perimeter is going to be 6 + 6, which is 12, plus another three, plus another three, so plus another six. So our new perimeter is going to be 18. And what's our new area? Our new area, or the area of the scaled rectangle, is 6 * 3, which is 18.

So what happened here? So our perimeter scaled up by the same as the scale factor. Our perimeter—our scale factor was three in this example—and our perimeter scaled up by three. But here, the area did something different. Our area scaled up by nine, and actually what it's scaling up by is 3 squared, which we know is 9.

Because if you increase in the, let's call it one dimension, by a scale factor, you're also doing it by the other dimension. And so if they're both scaled by three, when you do the area, you're going to be scaling it times three twice, or 3 squared.

So if we have a scale factor of three, perimeter is going to be scaled by three by three. So the new perimeter is going to be 33. Now area, as we just said, that's going to be scaled by a factor of nine. So if our new area is 72, our old area was 72 divided by 9, which is going to be equal to eight, and we are done.