Lesson video

Hi, my name's Mr. Chan.

And in this lesson, we're going to learn how to find areas of similar shapes given corresponding lengths.

Let's start with this example, we've got two rectangles, A and B, which were told are similar.

So that means that one rectangle is just an enlargement of the other rectangle.

So the first question is what is the length scale factor from A to B? So the length scale factor is just finding what we've multiplied by to get from one side length, two, from rectangle A to get through the rectangle B.

So we can see that we would multiply five by two to get to rectangle B's side length of 10 centimetres.

So the scale factor that would just be two.

Now we can work out the width of rectangle B by using the scale factor of two.

So we look for the corresponding side length in rectangle A.

So two multiplied by the scale factor two, would get me 4 centimetres for the width of B.

So now we can work out the area of each shape.

Rectangle A, length times width would be five times two to give me 10 centimetres squared, and rectangle B we would multiply the length and the width together to get 40 centimetres squared.

So now the question is, what is the area scale factor from A to B? So how many times bigger is the area of B than A? So 10 centimetres multiplied by a scale factor to get 40 centimetres.

Well, that's multiply by four.

So the area scale factor is four.

What do you notice? Here is another example, triangles A and B are similar, so we can see those in the diagrams there.

We can see B is just an enlargement of triangle A, and we can work out the length scale factor from A to B by looking at corresponding side lengths and seeing what it's multiplied by.

I'm looking at the base seven centimetres in A, 21 centimetres in B.

I can see that I would multiply by three to get from A to B.

So that would be the scale factor for the length.

Now I can work out the height of triangle B by using the same scale factor in the lengths for the heights of triangle B, two multiplied by three would get me to six centimetres.

So the height of triangle B is six centimetres.

Now using those heights and the base lengths, I can work out the area of each shape.

Remember, it's multiply the base and the height and half it.

So seven centimetres squared for triangle A, the area of B would be 21 multiply by six and halved, 63 centimetres squared.

So let's have a look at what the area scale factor is from A to B this time.

So what do I multiply the seven centimetres squared by to get to 63 centimetres squared? I would multiply that by nine, seven multiplied by nine gets me to 63 centimetres squared.

So the scale factor for the area is nine.

What do you notice? Here's a question for you to try.

Pause the video is complete the task, resume the video once you're finished.

Here are the answers.

So the examples should have helped you answer these questions.

What you find is, that in part A, you have a length scale factor of two, that means the area scale factor will be four.

That means the area of rectangle B will be four times the area of rectangle A.

So 15 multiplied by four equals 60 centimetres in part A, in part B, the area scale factor, let's have a look at the length scale factor first.

Well, the length scale factor is three, so the area scale factor will be nine.

So the area of rectangle B will be nine times bigger than the area of rectangle A.

So 24 multiplied by nine equals 216 centimetres squared.

So let's review the examples that we've looked at previously.

We had the two rectangles, which had a length scale factor of two, and we found the area scale factor to equal four.

In the triangles example, we had a length scale factor of three, and the area scale factor was nine.

What we should have noticed is that the area scale factor is the length scale factor multiplied by itself.

So if we look at the rectangles, who if the length scale factor was two, the area scale factor would be the length scale factor two multiplied by itself two multiply by two equals four.

So the length scale factoring the triangles example was three, three multiplied by itself, three times three would equal nine.

So that gave us the area scale factor and what we call multiplying a number by itself, which is simply squaring the number.

So let's apply that to another example.

If we've got two trapezia, which are similar, we can work out the length scale factor, and then the area scale factor.

So let's have a look at the length scale factor.

The length scale factor from trapezium A to trapezium B, five multiplied by six would equal 30 centimetres on the base of those trapeziums. So the length scale factor equals six.

Now the area scale factor is the length scale factor multiplied by itself.

It's the length scale factor squared.

So the length of the area scale factor would be six squared, which is 36.

So if we're told the area of trapezium A, for example, the area of trapezium A equals 24 centimetres squared.

We can use the area scale factor to work out the area of trapezium B, which would be the area of scale of trapezium A, 24, multiplied by the area scale factor, which is 36, 24 multiplied by 36, equals 864 centimetres squared.

So that is my area of trapezium B.

Here are some questions for you to try.

Pause the video to complete the task, resume the video once you're finished.

Here are the answers.

Let's look at question two.

If shape D is similar to shape C, we can work out the area of shape D by working out the length scale factor first.

So three centimetres to six centimetres, that means the length scale factor is two.

So three times two gets me to six, and that means the area scale factor will be the length scale factor squared, two squared, that means the area scale factor will equal four.

So the area of shape D is four times the area of ship C.

So four multiply by 25, that equals 100.

So the area of shape D is 100 centimetres squared.

Here's a question that you can try.

Pause the video to complete the task, resume the video once you're finished.

Here's the answer.

So what you can see in this question, is that Alex has thought that because the length scale factor is two, that means the area will also be double the size, and that's a very common mistake.

So make sure that you understand that that's not the case, and in fact, the area scale factor, is the length scale factor squared.

So in this example, if we know that the length scale factor is two, that means the area scale factor will be two squared, which is four Here is another question for you to try.

Pause the video to have a go at this, resume the video once you're finished.

Here's the answer.

My advice to you on this question would be to draw the two trapezia separately so that you can then work out which of the corresponding side lengths, and then work out the area that the question's asking you for.

Hopefully you did that and you got the correct answer.

That's all for this lesson.

Thanks for watching.