Lesson video

Hello, and thank you for choosing this lesson.

My name is Dr.

Rowlandson, and I'm excited to be helping you with your learning today.

Let's get started.

Welcome to today's lesson from the unit of 2D and 3D Shape with Compound Shapes.

This lesson is called Checking and Securing Understanding of Perimeter for Standard Shapes.

And by the end of today's lesson, we'll be able to calculate the perimeter of rectangles, triangles, parallelograms, and trapeziums efficiently.

Here are some previous keywords that we're going to use again during today's lesson.

So you might want to pause the video if you want to remind yourself what any of these words mean before pressing Play to continue.

This lesson contains three learning cycles.

And in the first learning cycle, we're going to focus on the side properties of different polygons.

Here we have five shapes.

What do these shapes have in common with each other? Pause the video while you think about this and press Play when you're ready to continue.

The main thing that these five shapes have in common with each other is that they all have four sides and they all have four vertices as well.

That means they're all quadrilaterals.

A quadrilateral means a four-sided polygon.

So let's take a look at some of these quadrilaterals together now.

Let's start off with these two here.

Yes, one thing they have in common is that they both have four sides, but these two quadrilaterals have more things in common with each other than that.

Pause the video while you think about that and press Play when you're ready to continue.

So what do these two four-sided shapes have in common with each other? Well, they are both rhombuses as they both have four sides which are all equal length.

And you can see that with both of these shapes because they have little hash marks on each of the sides.

Each side has a single hash mark.

That means those sides are equal in length to each other.

Now you might be thinking, "The shape on the left is not a rhombus, it's a square." And you'll be right in saying it's a square, but a square is a special type of rhombus, where the four angles are also all equal as well.

All the angles are 90 degrees.

With the shape on the left, you don't always have to say it's a square and a thrombus.

We tend to just say it's a square, but it's good to bear in mind that a square is a type of thrombus and a thrombus is a type of quadrilateral.

How about these two shapes? These are two different quadrilaterals.

And yes, they both have four sides, but they have more things in common with each other than that.

What else do they have in common with each other? Pause the video while you think about this and press Play when you're ready to continue.

Well, both of these quadrilaterals have two pairs of parallel sides that are of equal length to each other.

That means they are both called parallelograms. Now you might be thinking about the shape on the left is not a parallelogram, it's a rectangle.

And once again, you would be right with that, but a rectangle is a special type of parallelogram.

A parallelogram with four 90-degree angles is called a rectangle.

It's a bit like the relationship between a square and a thrombus.

A square is a special type of thrombus and a rectangle is a special type of parallelogram, and they are all quadrilaterals.

Now we can see that they have sides of equal length, and we can see that they have two pairs of sides are equal length because of the hash marks.

We can see that these two sides here have a single hash mark.

That means that those two sides are equal in length to each other.

And then these two sides have a double hash mark.

That means that those two sides are of equal length to each other.

So let's check what we've learned, and I've got a bit of a riddle for you.

I'm a quadrilateral with four sides of equal length and none of my angles are right angles.

What is the name of this quadrilateral? Pause the video while you write it down and press Play when you're ready for an answer.

This shape is a rhombus.

A rhombus has four equal sides, but it doesn't have four equal angles.

Otherwise, it would be a square.

Here we have a shape, which is a parallelogram.

You can see that two lengths are given to you, but two lengths are labelled with a and b.

And what you need to do is work out what are the lengths of the sides labelled a and b.

Pause the video while you do this and press Play when you're ready for an answer.

Well, here are your answers.

A is 12.

7 and the length is centimetres.

And we know that because it has one hash mark and so does the other length that has one hash mark as well, which is 12.

7 centimetres.

And b is 18.

3 centimetres because that length has double hash mark and so does the length that is parallel above it, which has 18.

3 centimetres.

So those are equal.

Let's think about trapeziums now.

Do trapeziums have sides of equal lengths? And we've got an example of a trapezium on the screen for you to see.

Pause the video while you think about this and press Play when you're ready to continue.

Well, trapezium can have sides that are of equal length to each other, but it doesn't have to.

Trapezium can have zero sides that are equal length, two sides that are of equal length, or three sides that are of equal length, but it doesn't have to have any of those in particular for it to be a trapezium.

It just needs to have a single pair of parallel sides.

Now that means it's especially important to pay attention to the hash marks on trapezium because that'll tell us which sides are equal in length to which other sides.

Let's take a look at some examples.

Here we have a trapezium that has no equal sides.

Here we have a trapezium that has two equal sides.

The left and the right side are equal to each other, and we can see that because they both have a single hash mark on them.

This particular type of trapezium is called an isosceles trapezium.

It is symmetrical.

In this case, down the vertical line that goes to the middle of each of it.

The left side of the trapezium is the same as the right side of the trapezium.

Here we have another trapezium that has two equal sides.

This time though, it's the top side and the side on the right.

This one is not an isosceles trapezium because it's not got that line of symmetry through the middle of it.

And here we have another trapezium.

This one has three equal sides.

The left, the top, and the right are all equal to each other because they all have that single hash mark.

And 'cause we have that line of symmetry going through the centre of it, vertically, it is isosceles trapezium again.

Here we have some other shapes.

What do these shapes have in common with each other? Pause the video while you think about this and press Play when you're ready to continue.

The main thing about these six shapes have in common with each other is they all have three sides and three vertices.

They're all three-sided polygons, which we call triangles.

Let's take a look at some groups of these triangles together now.

For example, here we have three triangles, which all have three sides, but they have something else in common with each other.

What do all three of these triangles have in common with each other? Pause the video while you think about that and press Play when you're ready to continue.

Well, the main thing that all three of these triangles have in common of each other is that they all have two sides that are of equal length, and that means they're called isosceles triangles.

They also have a line of symmetry that goes through them.

Now we can see that these have two sides of equal length based on the hash marks.

Each triangle has two sides with a single hash mark on it, which means that those sides are the same length as each other.

So how about these triangles? We have two groups of triangles here, one on the left and then two on the right.

What is the difference between these two groups of triangles? Pause the video while you think about this and press Play when you're ready to continue.

Well, the triangle on the left is an equilateral triangle.

That means it has three sides of equal length and the angles are all equal to each other as well.

Whereas the two triangle on the right, these are both scalene triangles.

These have no sides of equal length and the angles are all different as well.

Now one of those scalene triangles has a right angle.

This is a right-angle scalene triangle.

Now not all right-angle triangles are scalene.

We have a triangle also at the top there, which has a right angle, but that's is isosceles triangle.

So right-angle triangles can be is isosceles or scalene.

In this case, in the bottom box, we have a right-angled scalene triangle.

So let's check what we've learned.

Here we have four triangles and each triangle has one length on it that is labelled with a letter a, b, c, or d.

And on the right, you've got four answers, e, f, g, and h.

What you need to do is match the sides of each triangle to its missing length.

Pause the video while you work through this and press Play when you're ready for an answer.

Okay, let's take a look at some answers.

Here they are.

Here we have some more shapes now.

What do these shapes have in common with each other? Pause the video while you think about this and press Play when you're ready to continue.

Well, unlike the previous examples we've seen like this, it's not the case here that all these shapes have the same number of sides.

We can see we have a three-sided shape, a triangle, a four-sided shape, a quadrilateral, and a five-sided shape, a six-sided shape, and an eight-sided shape.

So what do they all have in common? Well, it's down to the hash marks and what we can see are the angles.

All of these shapes are regular polygons.

And that means that all the sides are of equal length and all the interior angles are the same within each shape.

And we can see that due to the hash marks and the angle markers.

For each shape, all the sides are labelled with a single hash mark.

That means that those sides are equal in length to each other.

And when it comes to the angles, well we can see for the square that all those angles are 90 degrees, so they are all equal to each other.

But for the other shapes, we can see that the vertices are all labelled with a single arc with no numbers given.

And that would suggest that those angles are all equal to each other within each shape.

So that means that the angles are all equal and also the lengths are all equal for each shape.

That means they are regular polygons.

Now, for some of those, we give special names too.

For example, an quadrilateral triangle and a square, but they are just examples of particular regular polygons.

We have a regular pentagon, regular hexagon, and regular octagon as well in these examples too.

So let's check what we've learned.

Here we have a regular hexagon, and think about what that means about each of the lengths of those sides.

So what is the sum of all the sides of this regular hexagon? Pause the video while you work this out and press Play when you're ready for an answer.

The answer is 120 centimetres because there are six lots of 20 centimetres.

Okay, it's over to you now for task A.

This task contains one question, and here it is.

You have seven shapes and each shape has at least one unknown length on it.

For each shape, label length of as many sides as possible.

And for each side where it's impossible to know the length of it, mark it with an X.

Pause the video while you work through this and press Play when you're ready for answers.

Okay, here are the answers.

Pause the video while you check these against your own and press Play when you're ready to continue with the lesson.

Fantastic work so far.

Now, let's move on to the second part of today's lesson, which is perimeters of polygons.

Here we have a quadrilateral.

How would you go about finding the perimeter of this quadrilateral? Before you think about that, here's a couple of suggestions.

Andeep says, "Just add up all the numbers that you see." And Aisha says, "Add up all of the distances around the edges of the shape." Whose method do you agree with here? Or are both these methods correct? Pause the video while you think about this and press Play when you're ready to continue.

Well, in this particular example, either Andeep's or Aisha's method will work in this case, but as we'll see a little bit later, Andeep's method will cause some problems for some examples, but we'll get to that shortly.

Let's start off with finding the perimeter of this shape.

The perimeter is the total distance around a shape, and that means the perimeter can be found by finding the sum of the lengths of every one of its sides.

So in this particular shape here, the perimeter can be calculated by adding together 8 + 9 + 16 + 12.

In this case, it would give 45.

And our units here are centimetres.

So the perimeter is 45 centimetres.

Here we have an example that looks very much like the one on the left, but what we can see here is that one of the lengths is unknown.

So when we try and find the perimeter here, we would do 4 centimetres + 5 centimetres + 11 centimetres +, oh, we don't really know how long that edge is.

So in this case we can't find the perimeter.

The perimeter cannot be found as all of its sides are not known.

So let's check what we've learned.

Here we have a polygon, and the statement says the perimeter of this polygon is 35 centimetres.

Is this statement true or is it false? Pause the video while you choose that and then press Play when you're ready for the second part of this question.

Well, the statement is false because the perimeter is not 35 centimetres.

So how can we justify that? Here are four possible justifications to choose from, and you may choose more than one.

Pause the video while you make a choice and press Play when you're ready for an answer.

the correct justifications are b and d.

It's not 35 centimetres because not all the size of the shape are known and because the real perimeter must be greater than 35 centimetres, 4 + 11 is 15, + 12 makes 27, + 8 makes 35.

And then there's another side to add onto that, which means that the perimeter must be greater than 35.

Here's another question.

You have three polygons.

And what you need to do is find the perimeter of each polygon.

But if it's impossible to find the perimeter of a polygon, mark it with an X.

Pause the video while you do this and press Play when you're ready for answers.

Well, shape A, the perimeter is 39 centimetres.

For shape B, it's not possible to find the perimeter because there's a length that is unknown.

And for shape C, the perimeter is 33 centimetres.

Here we have this time a six-sided polygon, a hexagon.

Andeep says, "Since I only know one of the six sides of this hexagon, I can't find its perimeter, right?" What do you think about that? Pause the video while you consider it and press Play when you're ready to continue.

Actually, you can find a perimeter since this is a regular hexagon.

We know it's a regular hexagon because there are hash marks on it that show that all the sides are the same length, which means each of those sides is 20 centimetres.

That means it is possible to find out the perimeter of a polygon if you can figure out the lengths of every side using other information such as hash marks, which indicate equal lengths, or being told it's a regular polygon before you start trying to find the perimeter of it.

So in this case, all the lengths are 20 centimetres.

So Andeep says, "I can label all six sides as 20 centimetres.

So its perimeter is 20 + 20 + 20 + 20 + 20 + 20, which is 120, and in this case, centimetres." Aisha listens in and she says, "That is correct! But can you also do 20 centimetres X 6 to get 120 centimetres?" And Aisha is correct here as well 'cause what we have is a repeated addition when we find a perimeter of this shape, and multiplication can help us with that.

So here we have three polygons, and all these polygons are regular.

What you need to do is pause the video and find the perimeter of each polygon and then press Play when you're ready to continue.

Okay, here are your three answers.

So here we have another polygon and two of its length are labelled.

And Andeep says, "Okay, you definitely can't find the perimeter of this shape since not all of its sides are labelled and it's not a regular shape like the previous one he saw." What do you think about that? Pause the video while you think about it and press Play when you're ready to continue.

Aisha says, "Of course you can, since it still has hash marks you can use to label of the sides." So let's take a look at those hash marks.

We can see the hash mark show that this shape has two pairs of sides of equal length.

There is one pair of sides with a single hash mark.

That means they are equal length.

And there's another pair of sides with a double hash mark, which means those are equal length.

So Andeep says, "Okay, lesson learnt.

I need to label as many sides as I can using the hash marks to help me before I know if I can find the perimeter or not." So in this case, we can see that our missing sides are eight centimetres and 20 centimetres.

So the perimeter will be the sum of all those sides, which is 56 centimetres.

Here we have another gramme, and Andeep says, "I cannot find the perimeter of this shape." What do you think about that? Is he right this time or not again? Pause the video while you think about it and press Play when you're ready to continue.

This time, Andeep is correct.

The double hash mark show that the bottom side is 35 centimetres because it's equal to the top side, but there is no way of knowing what these other two lengths are.

We know they're equal to each other, but we don't know what they're equal to.

So we can't find the perimeter.

Let's check what we've learned.

Here we have two polygons, they're both quadrilaterals.

You need to find the perimeter of each polygon.

But if it is impossible to find the perimeter of a polygon, mark it with an X.

Pause the video while you do that and press Play When you're ready for an answer.

Here are your answers.

Shape A, you cannot find the perimeter of that because you don't know the length of the left-hand side.

And for shape B, the perimeter is 56 centimetres.

Yes, the left and right sides aren't labelled, but those hash marks mean that it's equal to the 12 centimetres.

So you do know all the lengths.

Here we have another parallelogram, and Andeep says, "15 + 15 + 5 + 5 = 40.

So I know the perimeter is 40 centimetres." Is Andeep correct here? Pause the video while you think about this and press Play when you're ready to continue.

Well, Aisha says, "I don't think we can find the perimeter of this parallelogram." Let's take a look at why.

This five centimetres is here shows the vertical height of the gramme, how far it is from the bottom to the top of it in this case, but it's not actually the length of any of the sides on this shape.

So when we try and find the perimeter of the shape, we can do 15 +, well, we don't know how long this side is.

We can add another 15, but again, we don't know how long this side is.

So we cannot find the perimeter of this particular shape.

And this exemplifies a really important point when it comes to finding a perimeter because sometimes, some of the distances that you are given are not the lengths of sides.

They might be distances from one side to another, but not along one of the sides of the shape.

So it's important to identify which distances are sides of the shape and which one are not.

For example, here we have a quadrilateral, and we can see we have three different length given to us, but one of those is not the length of a side.

So when it comes to finding the perimeter, we can do 22 centimetres + 10 centimetres, but we can also + 10 centimetres and another 10 centimetres because those two sides have hash marks on them as does the first 10 centimetres.

So all three of those are equal to each other, and that gives 52 centimetres for our perimeter.

But what we can see here is that this eight centimetres was not used.

This eight centimetres shows the vertical height of the trapezium.

So it isn't involved in finding the perimeter of the shape.

It is not one of the lengths of the sides.

Now, you might think that every single time there is a shape with hash marks, you can find the perimeter.

Well, that's not the case.

Sometimes perimeters of shapes or hash marks cannot be found, such as with this one here.

We can do 18 centimetres + 11 centimetres, but we don't know how long this side is.

This top side isn't labelled with a length, nor does it have a hash mark to show its equal length to any other sides.

So we don't know how long that side is.

We'd plus number 11 to it and we wouldn't use the nine centimetres, but still the perimeter can not be found in this case.

So let's check what we've learned.

Here we have two shapes, two polygons.

Find the perimeter of each polygon.

If it is impossible to find a perimeter, mark it with an X.

Pause the video while you do this and press Play when you're ready for answers.

Here are your answers.

Shape A has a perimeter of 36 centimetres.

You did not use a 12 in that one, you do 10 + 13 + 13.

For shape B, you cannot find the perimeter of that.

Yes, you have that 16 centimetres, but it's not the length of any of the sides.

You don't know how long those sides are, which have a single hash mark.

That's why you can't find the perimeter.

Okay, it's over to you now for task B.

This task contains three questions, and here are questions one and two.

Pause the video while you work through 'em and press Play when you're ready for question three.

And here is question three.

Pause the video while you work through this and press Play when you're ready for some answers.

Okay, let's see how we got on.

Here are answers to question one.

Pause while you check against your own and press Play when you're ready for question two.

And here is question two.

Once again, pause while you check these answers against your own and press Play when you're ready for question three.

Here is question three.

Pause while you check against your own and press Play when you're ready for the third part of this lesson.

You're doing great.

Now let's move on to the third and final part of today's lesson, which is using perimeter to calculate missing lengths.

Here we have a quadrilateral, where we know the length of three of its sides, but one of its sides is unknown and labelled with an x.

And we know the perimeter of a shape, it is sometimes possible to construct and solve an algebraic equation to find the length of a missing side.

For example, if we know that the perimeter of this quadrilateral is 30 centimetres, then we can use this fact that perimeter equals the sum of all the side lengths to create an equation.

The perimeter is 30 centimetres.

So the left-hand side of my equation here would be 30 =, and the sum of all the side lengths is 4 + 5 + 11 + x.

So now I have an equation.

I can simplify this part of my equation to get 30 = 20 + x.

And then I can solve this equation by subtracting 20 from both sides to get x = 10 or 10 = x.

That means that the length there is 10 centimetres.

So let's check what we've learned.

Here we've got a pentagon where four of its side lengths are given, but one is unknown and labelled with a t.

But we do know that the perimeter of this pentagon is 50 centimetres.

On the right-hand side, you've got five equations labelled a to e.

And the question you need to answer is, which of these equations can be solved to calculate the length of the missing side labelled t centimetres? Pause the video while you work through this, and it might be more than one answer, and then press Play when you're ready for the answer.

Well, there are two correct answers here.

They are c and e.

Both of those equations can be simplified and solved to give the value of t.

So here's one of those equations.

Write down a simplified version of this equation and solve it to find the value of t.

Pause the video while you do this and press Play when you're ready for an answer.

Here are your answers.

The simplified version of this equation is 50 = t + 35.

You could also put 35 + t.

It doesn't matter which way round they go, but your answer is t = 15.

Here we have a pentagon this time, and we can see it's a regular pentagon because each of the sides is labelled of a hash mark, but we don't know the length of its sides and it's labelled with a p.

If we know the perimeter of a regular pentagon though, we can construct and solve an equation to find the length of one of its sides.

And we know one of its sides, we know all of its sides.

For example, if we know that the perimeter of this regular pentagon is 80 centimetres, well, a regular pentagon has five sides of equal lengths and we can, once again, use this fact to create our equation.

The perimeter is 80, and the sum of all the side lengths is p + p + p + p + p.

That can be simplified to get 80 = 5p.

And to solve it, we can divide both sides by 5 to get p = 16.

So that length must be 16 centimetres.

So let's check what we've learned.

Here we have a regular polygon.

Which of these equations can be solved to calculate the length of one side of this regular polygon? Pause the video while you work through this and press Play when you're ready for an answer.

The answer is e.

So what is the value of e? Pause the video while you work through this and press Play When you're ready for an answer.

The answer is 72 divided by 8 to give 9.

This time we have a rectangle, and what we can see here is one of its side lengths is labelled 11 centimetres, another side length is labelled y centimetres, which means it's unknown, and the other two side lamps aren't labelled of anything in particular, but they do have hash marks on, and that might help us along the way.

Because if we know the perimeter shape, it is sometimes possible to construct and solve an algebraic equation to find the length of multiple missing sides of equal length.

So let's do that together now with an example where the perimeter is 54 centimetres.

We could create our equation using this fact.

And here we'd have 54 = y + 11 + y again, 'cause that side is equal to the other side with a double hash mark, + 11 again because that side is equal to the other side with a single hash mark.

So now we've got an equation, 54 = y + 11 + y + 11.

That can be simplified.

We can simplify the two y's we have here to 2y.

We can simplify these two 11s to 22.

And then we have 32 = 2y, and we can solve it to get 16 = y or y = 16.

That means that both of these lengths are 16 centimetres.

So let's check what we've learned.

Here we have a triangle.

It's an isosceles triangle and it has perimeter 59 centimetres.

Which of these equations can be solved to calculate the length of one side of this polygon? Pause the video while you do this and press Play when you're ready for an answer.

D.

So what is the value of w? Pause the video while you get that and press Play when you're ready for an answer.

The value of w is 16.

Okay, it is over to you now for one last time for task C.

This task contains four questions, and here are questions one and two.

Pause the video while you work through this and press Play when you're ready for questions three and four.

And here are questions three and four.

Pause the video while you work through 'em and press Play when you're ready for answers.

Let's see how we got on.

Here are the answers to question one.

We have a is 15 and b is 7.

Question 2, d is 12 centimetres, c is 13, e is 14.

Question three, f is 12, g is 13, and h is 10.

And then question four, well, j is 37 and k is 42.

Wonderful work today.

Now let's summarise what we've learned during this lesson.

The main theme in this lesson has been about using the properties of polygons to work with perimeter.

We've learned that sides with equal lengths can be shown using hash marks, but it's not always possible to find the length of all sides of a polygon if the information isn't given to us or hash marks don't help us along the way.

The perimeter of a polygon is the sum of the lengths of all the sides of that polygon.

All the sides of a regular polygon are equal in length, so sometimes that can help us along the way.

But the lengths of a missing side of a polygon can also be found by constructing an algebraic equation if you know the perimeter of that polygon.

Great work today.

Thank you very much.