The following shape is made from 15 identical rectangles. Calculate the area of the shape.
So first of all, if we’re looking to solve this problem, we can see that the diagram is not drawn accurately. This is important because it means that we can’t just measure and then find out the width of the rectangle and therefore find out the area. Now, to work out the area of our rectangle, so that’s our total shape, it’s gonna be equal to the length multiplied by the width. So we know the length of our total shape, but we don’t know the width. And to find this out, what we’re needing to do is find out the length and width of each of our small identical rectangles.
So we’re gonna solve this problem in three parts. So we want to find the width and length of our smaller rectangles. So the first part we’re gonna do is find the length of one of our smaller rectangles. And we can do this because we can see that there are three smaller rectangles whose lengths added together is gonna make up the total length which is 14.4 centimetres. So the calculation we’re gonna do is that 𝐿 is equal to 14.4 divided by three. We do that because, as we said, the total length of the bigger rectangle is 14.4. There are three lengths of the smaller rectangles making this up. So therefore, if we divide 14.4 by three, we’ll find 𝐿. So when we do this, we calculate the answer 4.8 centimetres. So therefore, we know that 4.8 centimetres is the length of one of our smaller rectangles. So what’s the next stage?
Well, next, what we can do is we can work out the width. And we can work out the width because we can see that there are four widths make up the same length as 𝐿 which is our length. So therefore, we can say that the length is equal to four 𝑤, four multiplied by the width. So then, if we substitute in our value for 𝐿, which was 4.8 because it’s the length of one of our smaller rectangles, we get 4.8 is equal to four 𝑤. Well, four 𝑤 means four multiplied by 𝑤. So we’re gonna do the inverse instead of multiplying by four. We’re now gonna divide by four to find one 𝑤. And when we do that, we get 1.2 centimetres is equal to 𝑤. And that’s because 4.8 divided by four is 1.2. Okay, great, so now we have the length and the width of the smaller rectangles.
Well, now, what we want to do is solve the problem. And we want to find the area of the total shape. So how are we going to do this? Well, we have two ways that we can approach it. I’m gonna show you the first way and then another way to check. Well, we can see that the total width of the large rectangle is five 𝑤. Because it’s made up of five of the widths of the smaller rectangles. So therefore, we’ve now got the length and width of our larger rectangle. So therefore, we can say that the area is gonna be equal to 14.4 because that’s the length of the larger rectangle multiplied by five multiplied by 1.2. And that’s because that’s 1.2 is the width of the smaller rectangles. So therefore, this is gonna be equal to 14.4 multiplied by six. So this is gonna give us a final answer of 86.4 centimetres squared, not forgetting the units. And it’s centimetres squared because we’re dealing with an area.
Okay, great, so we found the answer. But like I said, I was gonna show you another method we could’ve used to work out this final part. And it’s also a good way of checking. Well, to find out the area of the small rectangle, we could work out this by multiplying the length by the width. So 4.8 was the length of one of our smaller rectangles multiplied by 1.2, which was the width of one of our smaller rectangles, which will be equal to 5.76. Then we know that there are 15 identical rectangles. So we want 15 of these. So therefore, to work out our total area, we’re gonna multiply 5.76 by 15, which once again gives us a final answer of 86.4 centimetres squared.