Lesson video

Hello, my name is Dr.

Rowlandson and I'll be guiding you through today's lesson.

Let's get started.

Welcome to today's lesson from the unit of angles.

This lesson is called Checking and Securing Understanding of Polygons, and by the end of today's lesson, we'll be able to recognise and name polygons with five or more sides and recognise the difference between regular and irregular polygons.

Here are some previous keywords that will be useful during today's lesson, so you may want to pause the video if you want to remind yourself what any of these words mean and then press play when you ready to continue.

This lesson contains two learning cycles and we're going to start with identifying regular and irregular polygons.

A polygon is a flat closed figure made up of straight line segments.

Let's take a look at some examples.

Here are some examples of polygons.

Some of these may look familiar to you, other ones not so much.

Let's consider why they are all polygons.

They're all flat in that they are all 2D, none of them are 3D shapes, they're all closed, which means that they don't have any gaps in them.

If you imagine each one being a field with a fence around it, then there is no way for an animal in that field to escape out of a gap between the fences, or you can even think of it as if you start off in one vertex and walk around each of the sides, you will eventually end up back where you started.

And they're all made up of straight line segments, there are no curves in any of these shapes or wibbly wobbly lines in any kinda way.

These are examples of shapes that are not polygons.

Can you think about why each of these is not a polygon? Pause the video while you think about that and then press play when you're ready to continue.

Well, let's take a look at 'em together.

The shape in the top left of this set of examples looks like a circle, that's not a polygon because it's not made of straight line segments, and the same as well with the shape in the top right.

It has two straight line segments, but one of them is curved.

The shape in the bottom left, that one has got made by straight line segments, but there is a gap in it so that it is not closed.

That's why it's not a polygon.

And the shape in the bottom right, that is a cylinder, it's a 3D shape, so that one is also not a polygon because it's not flat.

The word, polygon, is derived from ancient Greek for many angles.

Polygons are named according to how many angles or sides they have.

For example, we have an octagon and a decagon.

You can see the names of these two shapes are made from two parts.

We have oct or dec, and then the second part is agon for both of them.

Those two parts indicate what this shape is.

An octagon has eight angles, oct being eight and agon being angles, a decagon has ten angles, dec meaning ten and agon meaning angles.

And here are some other examples of the names of polygons, which are based on how many angles they have.

Certain properties of a polygon can be indicated by including specific markings on the diagram.

Let's take a look at some examples.

Hash marks show sides which are of the same length.

So if two sides or more than two sides have the same number of hash marks, they are of equal length.

In this case we can see that the side on the left is equal to one the right, they both have one hash mark, and the side at the top is equal to the side at the bottom because they both have two hash marks.

Feathers or arrows, show sides which are parallel to each other.

If two or more sides have the same number of arrows, they are parallel to each other.

In this case, we can see the side on the left and the side on the right both have one arrow, so they're parallel to each other, and the side at the top and the side at the bottom have two arrows, so they are parallel to each other.

And angle markers show angles which are the same size.

If they have the same number of arcs, they are equal to each other unless there's any other information to indicate otherwise.

That usually happens when angles are just labelled with one arc.

It might be, but if labelled with one arc and they have different numbers on, that's fine, but if there's no numbers and you can see that there are different numbers of arcs angles, it usually indicates that they are equal size to each other.

Equal sized angles can also be shown using algebra.

So for example, the top side and bottom side are both labelled with a length A, because they both have the same letter, they're both at the same length.

When it comes to the angles, you can see that there are two angles labelled X degrees use same letter, it means that they're the same size angle.

Let's think now about regular and irregular polygons.

Regular polygons have sides that are all equal in their length and interior angles that are all equal in their size.

For example, here we have three regular polygons.

We have an equilateral triangle, a square, and a regular pentagon.

Irregular polygons have size that are not all equal and, or interior angles that are not all equal.

And here are some examples.

We have an isosceles triangle where two of its sides are equal, but the other one is not and that's why it's irregular.

Same can be said for its angles as well.

We have a rectangle where all the angles are equal, but not all the sides are equal, so that's why it's irregular.

And we have an irregular pentagon where some of the sides are equal to other sides, some angles are equal to other angles, but because all the angles are equal to each other and because all the sides aren't equal to each other, it's an irregular pentagon.

So let's check what we've learned.

Which of these polygons is regular? Pause the video while you choose and press play when you are ready for an answer.

The answer is A.

A is regular because all the angles are 120 degrees and all the lengths are four units.

So let's look at this particular polygon.

Why is this polygon irregular? You've got options A, B, and C to choose from.

Pause while you choose and press play when you're ready for an answer.

The answer is B, not all of its sides are the same length.

We can see all its angles are, but not all the sides.

That makes it irregular.

What about this polygon? Why is this one irregular? Pause why you choose from A, B, or C and then press play when you're ready for an answer.

The answer is A, not all of its interior angles are the same size.

We can see all the sides are the same length, but some of the angles are 141 degrees while other angles are 78 degrees, and because they're not all equal, it means it's irregular.

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

This task contains two questions and here is question one.

You have four quadrilaterals and you have a table each labelled A, B, C, and D, and there are properties of quadrilaterals in the headings for that table.

What you need to do for each quadrilateral is tick the appropriate boxes in the table.

Pause the video while you do that and press play when you are ready for question two.

And here is question two.

You have six shapes and you've got a table.

You need to sort the six shapes into the table to indicate whether it's a regular polygon, an irregular polygon or not a polygon, and you may choose to measure parts of the shape to help you as well.

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

Okay, let's now go over some answers.

Question one, shape A, all the angles are not equal, all the sides are not equal, that means it's not regular, it's irregular shape.

Shape B, all the angles are not equal, but all the sides are equal, either way, it's not regular, it's irregular because all the angles are not equal.

For C, we can see that all the angles are equal, but all the sides are not equal.

That means it's not regular, it's irregular.

And for shape D, all the angles are equal, all the sides are equal, and because we've ticked both of those, that means it is regular.

Then question two, we need to sort the shapes into the correct columns of the table.

Shape A, well, that one is not a polygon, that's because it is not made of straight line segments.

Shape B, it is a regular polygon.

It's a polygon because it's a flat closed shape made of straight line segments, and if you measure those lengths, you'll see that they're all equal size and if you measure your angles, you'll see that they're all equal size as well, it's a regular octagon.

Shape C, that one is a polygon, but it's an irregular polygon.

Not all the side lengths are the same for the shape C, the angles are but not the side lengths.

For D, that one is a regular polygon.

If you measure the sides and measure the angles, you'll see that they're all equal and it is a flat closed polygon, it's an equilateral triangle.

E is an irregular polygon.

Again, it's a triangle, but it's not equilateral.

And F, that's not her polygon.

It's closed, it's flat, but it's not made of straight line segments.

Fantastic work so far, now let's move on to the second part of this lesson, which is finding unknown measurements in regular polygon.

If you know that a polygon is regular, then you can use the length of one side to find its perimeter.

For example, here we have a regular polygon on the left where you know one of the sides is five centimetres and you know all the sides are five centimetres.

So that one you can work out the perimeter.

The perimeter can be found by using this information by doing five multiplied by the number of sides.

But with the example on the right, that one is not regular.

You know what the bottom length is five, but you don't know what any of the other lengths are, which means more information is required before the perimeter can be found.

So let's do an example of this together and then you will have one to try yourself.

Here we have an heptagon.

Again, it has seven sides.

The heptagon again below is regular.

Find its perimeter.

Well, we can see that one of its sides is five centimetres and we know it's regular, which means all the sides are five centimetres.

There are seven sides, so we can do 5 times 7 to get 35 centimetres.

Here's one you to try.

You've got a pentagon, it is regular.

Find its perimeter.

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

The answer is 35 centimetres again, which you do by doing seven times five.

But the seven this time is the length of each side and five is the number of sides.

Either way, these two shapes have the same perimeter.

Here's another example.

The heptagon below is regular, and you are told that its perimeter is 21 centimetres.

And what we need to do is find the value of X.

Well, we know that it's regular, which means if that side is X length centimetres, then all the sides have a length of X centimetres, which means you have seven lots of X, which make up that 21.

We could then do 21 divided by 7 to get a length of 3 centimetres, which means the value of X is 3.

Here's one for you to try.

The pentagon is regular, its perimeter is 45 centimetres and you need to find the value of X.

Pause the video while you do this and press play when you are ready for an answer.

The answer is nine.

You get that from dividing 45 by 5.

Here's another question to check your understanding.

For which hexagon or hexagons, is it possible to calculate the perimeter using the information provided? Pause video while you choose and press play when you're ready for an answer.

The answer is A.

In that question, you know it's regular, so you can work out that all the lengths are four centimetres and calculate its perimeter.

With B, you know all the angles are equal, but you don't necessarily know that all the sides are equal.

You could take that hexagon and stretch it in a horizontal way, which would keep all the angles the same, but it would adjust the length of the top side and bottom side.

So we don't necessarily know that it is going to be regular.

And for C, well, you don't know the angles the same, you don't know the size of the same, we don't know it's regular.

So the fact we're working with here is that all interior angles in a regular polygon are equal and this fact can be combined with other angle facts to solve missing angle problems. For example, here we have a regular hexagon with a line going through it or a line segment.

We can see that that line segment is parallel to the top side and bottom side of this hexagon.

We're given one angle, we to work out the value of X.

How could we go about doing that? Pause the video while you think about this and press play when you're ready to continue together.

Well, there are different ways we can do this.

One solution could be to first find that the angle on the inside is 120 degrees because angles around a point sum to 360 degrees, and 360, subtract 240 is 120.

Once we know that, we know that this angle is also 120 degrees because all interior angles in a regular polygon are equal.

So how can we go about finding the value of X from here? Well, we know that this angle here is 60 degrees because co-interior angles in parallel lines sum to 180 degrees and we know that that line segment that goes from the middle of the hexagon and its top edge are parallel.

Therefore, the 120 degrees and the 6 degrees are co-interior between parallel lines.

And then we could do 120 degrees, which is the interior angle of the hexagon at that point, subtract the 6 degrees we just worked out and we get a value of 60 degrees for X.

Another way we could do it is like this.

We know that this is 120 degrees for the same reason again.

We know about that angle there which had the X on it as part of it is 120 degrees because all interior angles in a regular polygon are equal.

And then we know that this X here is 60 degrees because a regular hexagon is symmetrical.

So that line which we've drawn there by (indistinct) the angle, and therefore, 120 divided by 2 is 60.

So let's check what we've learned.

Here you've got a diagram, please find the value of X.

Is it A, 54, B, 72, C 90, or is it D, 144? Pause video while you make a choice and press play when you're ready for an answer.

The answer is A, 54.

We can get that because we know all those interior angles must be the same, must all be 144 degrees.

We can see that part of that interior angle with the X is 90 degrees, so we subtract 90 from 144, we get 54.

Here we have a slightly adapted version of this diagram.

What is the value of Y now? Justify your answer with reasoning.

Pause video while you do this and press play when you are ready for an answer.

The answer is 54, and here's an example of some reasoning.

The line segments indicated are parallel, which means alternate angles in parallel lines are equal and that Y degree's angle is alternate with the 54 degree angle we worked out earlier.

Okay, it's now for Task B.

This task contains four questions and here is question one.

You have a bunch of polygons, and all of these polygons are regular, but they are not drawn accurately.

All the lengths are given in the same unit and you can assume that.

And what you need to do is match the polygons that have the same perimeters.

Pause video while you do this and press play when you're ready for more questions.

Here are some more questions.

In question two, you are told a regular decagon has a perimeter of eight centimetres and you need to work out the length of each side.

In question three, you are presented with a heptagon, which has seven sides and you're told that its perimeter is 14.

8 metres, and what you need to decide based on the information you're given is, is it regular and justify your answer with reasoning, explain how you know.

And question four, you're given a diagram that contains a regular octagon and what you need to do is find the value of X in that diagram, justifying your answer with reasoning.

Pause the video while you do this and press play when you're ready to go through some answers.

Okay, let's go through some answers.

Here are your answers to question one.

A and G both have the same perimeter, they both have a perimeter of 36 units.

B and F have the same perimeter, they both have a perimeter of 40 units.

C and E have the same perimeter, they both have a perimeter of 42 units.

And then D and H have the same perimeter, the perimeter is 48 units.

And then with question two, a regular decagon has a perimeter of eight centimetres.

What is the length of each side? Length of each side is not 0.

8 centimetres.

You do it by divided eight by the number of sides which is ten.

Question three, the perimeter of this heptagon is 14.

8.

Is it regular? Well, no, it's not regular.

You can work it out by subtracting the six lots of 2.

1 that you have from 14.

8 and that'll give you that the remaining edge is 2.

2 metres long.

So that means that length is not the same as others, so it's not regular, it's irregular.

Now question four, to find the value of X and justify your answer with reasoning, X is 112.

5 and there are different ways you can get your answer.

One example answer is that the line segments indicated are parallel and corresponded angles on parallel lines are equal.

So that angle that now is appeared next to the X must be equal to angle you are given, both 67.

5, and the angles form a straight line sum to 180 degrees.

So the 67.

5 plus X must make 180, so X must be 112.

5.

Fantastic work today, now let's summarise what we've learned during this lesson.

A polygon is a flat or 2D closed figure made up of straight line segments.

If it's got curves in it, it's not a polygon.

The prefix poly is derived from ancient Greek and poly means many.

The agon part in polygon means angles, so that's why all the names of lots of names of polygons are something agon like octagon, eight angles, nonagon again, nine angles, and decagon, ten angles.

The prefix oct, non, and dec tell you how many angles it has, eight, nine, and ten, and the agon part tells you it's a polygon.

Regular polygons have sides of equal length and regular polygons also have interior angles of equal sides.

However, if a polygon has just one of those two properties, it doesn't necessarily make it regular.

If a polygon has equal length sides but not equal angles, it's not regular, it's irregular.

Or if a polygon has equal interior angles but not equal length of sides, it's not regular, it's irregular 'cause irregular polygons have at least one side with a different length, and, or at least one angle of a different size.

Great work today, hope you have a nice day.