Lesson video

Hi, welcome to today's lesson on calculating missing lengths with the perimeter of polygons.

By the end of today's lesson, you'll be able to use properties of a range of polygons and the perimeter of a given polygon to deduce any missing side lengths.

There are two sections to today's lesson and we're going to start with section one on calculating unknown lengths.

Here we have a rectangle.

We know that one of the lengths is six centimetres long but what about the other? Well, unknown lengths can be calculated if we know the perimeter and the properties of a polygon.

The perimeter of this rectangle is 42 centimetres.

How might you work out the unknown length? Pause the video while you write down what you think you might want to do.

Let's see if what you came up with is what I'm suggesting.

I know the perimeter, in other words the distance around this rectangle is 42 centimetres.

Now Alex's approach is to say six lots of W therefore so six times w must be 42.

That means the missing length.

I can work it out by doing 42 divided by six.

So using my knowledge of inverse operations that must mean that that missing length is seven.

Jacob has a slightly different approach.

Jacob says if I add W plus six, plus w plus six, so add that altogether that's the perimeter.

So 42.

I can gather the like terms. There's two lots of W there, and six and six is 12.

So 2w plus 12 must also be equal to 42.

I can subtract 12, and that tells me that two 2W must be equal to 30 and by dividing by two W must be 15.

Now, Izzy's approach is much shorter.

Izzy reckons that W plus six must be equal to 21.

Well, what do I add to six to make 21? 15.

So W must be 15, which is what Jacob said but it's a very different approach.

Pause the video and write down what you think of each person's approach.

What did did you say about Alex's approach? Did you spot? This is not how we calculate perimeter.

Alex has just worked out what that length would be if 42 was not the perimeter, but was in fact the area.

Oops, Alex has made a mistake.

Let's now look at Jacob and Izzy's working.

Both approaches got us to the same answer.

You might have said that you like Izzy's better because it's much shorter or more efficient.

Izzy's gone for the idea that the length plus the width, or in other words, the long side plus the short side of a rectangle is equal to half of the perimeter.

And in doing so, she's had a lot less to work out.

But both approaches work, and that's the key idea here, is that it didn't matter which approach I took I was always gonna get to the right answer.

Now, quick time for reflection.

To work out an unknown length in a regular polygon I just need the perimeter.

Is that true or is that false? Once you've made your selection don't forget to justify your answer by either picking A or B.

Pause the video now while you make your choice.

Did you spot the question referred to a regular polygon? What do we know about regular polygons? And that's right.

This particular statement is true because all the sides of a polygon that's regular are the same length so you can divide by the number of sides.

Here's an example to illustrate that.

The perimeter of this heptagon is 112 millimetres.

What's the length of one of its sides? Pause the video while you work this out.

A heptagon has seven sides.

We can see that this heptagon is a regular polygon because the markings are there to indicate all sides are the same length.

This tells us that the perimeter is equal to seven times one of the side lengths.

In other words, 112 is equal to seven times a side length.

So if we divide by seven we know that 16 is the length of one side.

It's now time for our first task,.

I'd like you to calculate the lengths of the marked sides.

So in question A, tell me what length side A has.

In B, tell me the length of side B.

Remember to use your knowledge of the markings that indicate which sides are the same length and look at the total perimeter.

Be careful because not all sides are necessarily the same length.

Pause the video while you work out A and B.

Welcome back.

It's now time to look at the final part of the first task.

In this shape, in other words a really large W, there are lots of sides that are the same length and other sides that are different but still grouped to be the same.

So there's lots of notation on here.

Be careful and make sure you know what each side length is.

The perimeter of the entire shape is 74 metres.

Work out what the length C is.

Pause the video now while you have a go.

Welcome back.

Let's go through the solutions.

In a we have a rhombus.

The rhombus means that all of its sides are the same length and we can see that with our notation.

Since the perimeter is 10, I know that a plus a plus a plus a or four A is equal to 10.

10 divided by four gives us 2.

5 or two and a half.

So the length for A must be two and a half units.

In B, we had some lengths that were the same and other lengths that were the same.

We actually had two groups here.

We've got three sides each of seven units long.

So seven, add seven, add seven or three lots of seven and we have two sides that have length B.

So we can see we've got B, add B or we could write that as 2B.

So the first two lines there are saying the same thing just in different ways.

Summing all the sides gives us 25 and we can write that as 21 plus 2B equals 25.

What do I add to 21 to make 25? Well, it must be four.

So 2B must be the same as four.

Well, if two lots of B is four, then B must be two.

So my missing length is two units long.

Now C was a big one.

I had lots of sides that were the same length.

Let's start by looking at the length of four metres and that's marked at the bottom.

We can see that all sides that are crossed by one line have length four.

There are five sides that therefore have length four.

So that's the first part of my working is five lots of four.

I'm then going to look the other side that I know which is the nine metre line.

Now that length is denoted by three lines that cross it and there's one other side that matches.

So there are two lots or two sides of nine metres length.

Then let's look at C, and there are six sides that have the same length.

So that's six lots of whatever length C is.

Adding that together gives us a total of 74.

Well, let's work out what we do know.

Five lots of four is 20 and two lots of nine is 18.

The 6C stays as it is for now and our total is still 74.

20 add 18 is 38 and 38 add 6C is 74.

Therefore 38 adds something is 74.

That something's gonna be 36.

So six lots of C is 36.

In other words, what do you multiply six by to make 36? And the answer is six.

So C has a length of six metres.

Well done if you've got all of that.

Particularly that last one.

There was a lot of notation there.

You may have found it easier to copy that diagram and label the size yourself so that it was easy to see.

If you had a printed copy of this diagram then perhaps you wrote all over it and that's absolutely fine too.

You need to do whatever makes it easiest for you to visualise the problem.

It's now time to look at the second part of our lesson.

This time we're gonna be looking at problems within a context.

It's important to remember that math is not just for the classroom.

Math is everywhere.

It's out there in the wide world and people use it in a variety of jobs even ones you might not necessarily think of.

Let's have a look.

Farmer Dylan is creating a rectangular paddock for their sheep, and you can see the paddock here.

So a paddock is just a fenced area.

The paddock needs a gate.

Well, of course it does.

How are our sheep gonna get in and out if we don't have a gate? Unfortunately, farmer Dylan forgot to measure the gap in the fence.

In other words he doesn't know how long his gate needs to be.

It's really important he gets this right.

If his gate is too long it's not gonna fit in the gap, and if it's too short well it's not gonna function very well as a gate, is it? Sheep are just gonna be able to walk out through the gap.

What I'd like you to do is list the different ways that you could work out what the length of that gait could be.

What information would you need to know? What about if you didn't have that information? Is there other information that would work? Pause the video while you reflect and write down what it is you need to know to be able to calculate the length of that gate.

Let's have a look at all the different ways you could calculate the length of the gate.

You could have written so many different things.

Here's just a couple of examples.

You could have said I want to measure the gap in that fence.

If I've measured the gap then I know the length of the gate.

You could have said, well, if I know the perimeter of the whole paddock and I know the length of the fence then I could just see what the difference is between them.

That missing amount must be my gap.

There are lots of other things you might have come up with too.

These are just the ones that I thought were the most likely but you may have surprised me.

Ah, now you've got some more information.

Farmer Dylan knows the paddock has a length of 90 and a width of 45.

In other words, the whole rectangle.

So if he considers everything, the entire length is 90, the entire width is 45.

Using that information, what are you able to calculate? Pause the video while you write down what you can work out and then have a go at working it out so you've got an actual value.

Pause the video now.

Using the information you can calculate the perimeter of the whole paddock.

You can either have done 90, add 45, and then double it or two lots of 90 add two lots of 45 or even 90, add 90 plus 45 plus 45.

In total, the perimeter of the paddock is 270 metres.

Now that's helpful.

If we know the perimeter of the rectangle.

If I just knew the length of the fencing used I could see what the differences between them and that would give me the length of the gate.

Ah, look, farmer Dylan knows that the total length of fencing they used was 195 metres.

Now, I can use my answer from before my perimeter of my paddock and the length of the fencing I've used to work out what the length of the gate must be that I need.

Pause the video now and work out what the length of the gate is.

The perimeter of the paddock was 270 metres.

The length of the fencing used is 195.

So the difference between these two is 75 metres.

That's how long my gate is going to need to be.

When dealing with a context-based problem, it can be beneficial to think about what you could work out if you had the relevant information.

So for example, in the problem we just looked at rather than going straight to the question we paused and we thought about it.

What could we work out? If we had this information, what could we do? And when we got a little bit of information we worked out what we could.

Perimeter calculations are not only used for a context involving sheep.

There are many different contexts in which you may need to work with the perimeter of a shape.

For example, the skirting board around a floor.

If you look at the room, you'll see there's often a border along the floor where the wall meets the floor.

The border that appears there is called a skirting board but you'll notice that it has to stop when we get to the door.

It can't go across that gap.

So a similar situation to earlier would be if we know the perimeter of our room and we know the length of our skirting board.

That's similar to knowing the perimeter of the paddock and the length of the fence.

So same maths would be involved even though we have a different context.

It's now time for our second task.

For this task, I'd like you to write your own problem using the same measures but in a different context.

So if you remember our sheep in our paddock we said that the perimeter of the paddock was 270 metres and we got that because the length was 90 and the width was 45.

I then asked for you to calculate that missing gap because I told you what the length of the fence was.

We then in our lesson said, well those same numbers could have applied to a different context.

For example, measuring around a room and we talked about the skirting board and this is what I'd like you to do.

Can you use those numbers and have them stand for the same thing, but put a different context on it? In other words, we're not talking about sheep and paddocks and we're not talking about rooms with skirting boards.

Can you think of another context where this could work? In other words, where you know the perimeter of the shape, you know its length and width.

So again, we must be talking about a rectangle here, length and width, and there's some sort of missing length.

So perhaps something only goes part of the way round it and you need to calculate that missing gap.

Now, this can be quite tricky.

So if need be why not try writing out my problem with the skirting board and applying that context to see that it still works in the same way as the sheep with the paddock first before trying to work out your own context.

Pause the video now while you have a go.

Welcome back.

You could have had lots of different answers for this.

The important bit is that you know there's a perimeter of 270 metres.

There was that known length of 195 in my question.

You might have picked a different total known length and that's okay if you did but you might have used the one that I did.

And then there was that unknown length you had to work out which in my case would've been 75.

What would it have been for you? If you picked exactly the same numbers as me you would've ended up with the same thing.

What context did you come up with? Did you go with my skirting build from earlier or did you pick something different? You might have gone for a rectangular picture frame and then at the bottom of the frame rather than having the frame go the whole way round you wanted a place maybe for a sign or a plaque.

You could have talked about therefore framing something or maybe a base for something to go on.

My particular favourite one would've been let's say you've baked a cake and you want to put a ribbon around the cake with a gap for some icing to go in.

How long's that gap got to be? Now, because we're talking about units involving metres that would've been a really big cake but you might have picked different units and just kept the numbers the same, and that's absolutely fine if you did.

There were quite a lot you could do here.

But if this is something that's not familiar to you it might have been really quite hard.

The point of the task is to realise that maths is all around us.

The real world is saturated with maths.

We just don't necessarily always see it and this task was designed to get you to have an opportunity to think about this particular bit of maths and all the different ways we could see it in the real world.

Let's summarise our lesson today.

We can calculate unknown lengths using the perimeter and any properties of a polygon.

It can be helpful to consider what you could work out if you have particular information for a given problem.

If you found the problem solving part at the end, quite tricky, there's no need to worry.

We're actually gonna do quite a few of those throughout this unit, throughout all of your lessons, not just this particular topic.

By the end, you'll have had quite a lot of practise on how maths is used in our real world.

I look forward to seeing you for our next lesson.