Lesson video

Hello, my name is Dr.

Rowlandson, and I'll be helping you with your learning during today's lesson.

Let's get started.

Welcome to today's lesson from the unit of Geometrical Properties with Polygons.

This lesson is called Deriving the Sum of Interior Angles in Multiple Ways.

And by the end of today's lesson, we'll be able to use reasoning to derive the sum of interior angles in lots and lots of different ways.

Here are some previous keywords that you may be familiar with already, and we'll be reusing again in today's lesson.

So you might wanna pause the video while you refamiliarize yourselves with these words before pressing play to continue.

This lesson contains two learn cycles, and it's all about trying to find a sum of interior angles in lots of different ways.

In the first learn cycle, we'll be exploring multiple different ways to split a polygon in order to derive the sum of interior angles, and in the second learn cycle, we're gonna use sequences to derive the sum of interior angles.

But let's start off with exploring multiple ways to split a polygon.

Here we have Izzy and Alex.

Izzy and Alex are finding the sum of interior angles of a pentagon by splitting it in different ways.

Here's Izzy's method.

She says, "I've split the pentagon into three triangles.

"Three times 180 is 540.

"So the interior angles of the pentagon must sum to 540." And here's Alex's method.

Alex says, "I've split the pentagon into five triangles.

"Five times 180 is 900, "so the interior angles of a pentagon must sum "to 900 degrees." So Izzy and Alex have different methods and have different answers, but who is correct, and what is wrong with the other person's reasoning? Pause the video while you think about this and then press play when you're ready to continue.

Hopefully, we've decided that Izzy's method is correct and angles in a pentagon sum to 540.

But what is wrong with Alex's reasoning? Well, the issue with Alex's reasoning is that he has got line segments that cross inside his pentagon.

When we draw line segments inside a pentagon that intersect each other, it creates more triangles, but it also creates an extra vertex inside the pentagon and that creates extra angles.

However, the additional angles are not at any of the vertices on the pentagon.

Therefore, the sum of the interior angles of the pentagon is not equal to the sum of the angles in the five triangles.

Now, while this method has produced a wrong final answer in this case, it's not a complete disaster.

Because with a little bit of tweaking, we can still use this method to derive the sum of the interior angles of a pentagon.

We just need to think a bit more carefully about what extra vertex has been created.

The new vertex has angles around the point which sum to 360 degrees.

And because that 360 degrees does not contribute to the sum of the interior angles of a pentagon, that is why the answer in this case is too big.

This issue can be rectified by subtracting 360 degrees for each extra vertex where all the angles around the point are not part of the interior angles of the polygon.

For example, in this case, to get from 900, which is our incorrect answer, to 540, which is the correct answer, we can just subtract 360 degrees.

And that is subtracting the extra angles created by that vertex inside the pentagon where those lines cross.

Let's take a look at a similar issue with a different shape.

Aisha is finding the sum of the interior angles of a hexagon.

She splits the hexagon into 10 triangles, and we can see what she's done so far.

Aisha's working so far, says, "10 times 180 degrees is 1,800 degrees." Aisha says that the interior angles of the 10 triangles sum to 1,800 degrees.

But she doesn't say that the interior angles of a hexagon is 1,800.

Can you think why? The reason why is because we have these extra vertices.

Aisha says, "The line segments "that I have drawn have created three extra vertices, "and these vertices are inside the hexagon, "and the angles in there do not contribute "to the interior angles of the hexagon." Aisha says, "The angles around the point "at each vertex add an extra 360 degrees to my total." So what she's saying there is that her total so far of 1,800 is greater than it should be, and she reckons it is 1,080 degrees greater than it should be, because it's got three lots of 360 adding to it.

So she says, "I can find the sum of interior angles "by subtracting three lots of 360 "from 1,800 degrees." She's done 1,800 degrees, subtract 1,080 degrees to get 720 degrees, and that is the sum of the interior angles of a hexagon.

Let's check what we've learned so far with this, by solving a problem in multiple stages.

Here we have Lucas.

Lucas is finding the sum of the interior angles of an octagon.

Octagon has eight sides.

He splits the octagon into 10 triangles, like we can see on the screen here.

What do the interior angles of the 10 triangles sum to? Pause the video while you write down your answer to this, and then press play when you're ready to continue.

The interior angles of these 10 triangles sum to 10 lots of 180, which is 1,800 degrees.

In Lucas's method so far, how many extra vertices has Lucas created? We're thinking about vertices where the angles do not contribute to the interior angles of the octagon.

Pause the video while you write down how many extra vertices Lucas has created and then press play when you're ready to continue.

Lucas has created two extra vertices, and because these vertices are in the middle of the octagon, the extra angles created by these vertices do not contribute to the interior angles of the octagon.

So our next question is, what is the sum of the extra angles that Lucas has created? Pause the video while you have a go at this, then press play when you're ready to continue.

The extra angles that Lucas created sum to 720 degrees.

So let's now solve the overall problem.

What is the sum of the interior angles in an octagon? Pause the video while you work through this and press play when you're ready for an answer.

Well, if the sum of the interior angles of the triangle is 1,800, and we recognise that that's too much because we have these two extra vertices in the middle, which each contribute 360 degrees, we just need to do a subtraction now.

If we do 1,800, subtract 720, we get the sum of the interior angles of the octagon is 1,080.

So earlier, we saw Izzy's method for how to split a Pentagon into triangles, and we can see that again on the left of the screen.

Now we have Sofia.

Sofia finds the sum of interior angles of a pentagon by splitting it into four triangles.

She compares her method now with Izzy's.

Sofia says, "My answer is 180 degrees greater than Izzy's answer." That's because she's got 720 and Izzy's got 540.

Why is Sofia's answer 180 degrees too big? Pause the video while you take a look at what Sofia's done, and then press play when you're ready to continue.

Well, Sofia's method looks quite similar to Izzy's.

She's started from one particular point and drawn line segments from that point to vertices on the pentagon, but can we spot the slight difference? In Izzy's method, she's started at a vertex and drawn line segments to each other vertex, but in Sofia's method, she's started partway along one of the edges and drawn line segments from there.

So that's why Sofia has four triangles rather than three.

But why has that made it 180 degrees too big? Well, when line segments intersect a polygon partway along an edge, it creates, again, an extra vertex.

Once again, the extra vertex creates additional angles, and these additional angles are not at any of the vertices of the polygon.

Therefore, the sum of the interior angles of the pentagon is not equal to the sum of the angles in the four triangles.

So let's take a look at this extra vertex.

Here it is.

It's created these four angles here, and what do you think these four angles in this vertex add up to? The new vertex has adjacent angles on a straight line which sum to 180 degrees.

And that is why this answer is 180 degrees too big, because those four angles, which add up to 180, do not contribute to the interior angles on the pentagon, they are extra.

So this issue can be rectified by subtracting 180 degrees for each vertex where there are adjacent angles on a straight line that are not part of the interior angles of the polygon.

We can do 720 degrees, subtract 180 to get 540 degrees.

Let's check what we've learned there.

Jacob is finding a sum of interior angles in a hexagon.

He splits the hexagon into six triangles.

How many extra vertices has Jacob created by doing this? Pause the video while you write down an answer and press play when you're ready to continue.

Jacob has created two extra vertices.

So let's use that now.

Here's what Jacob has done so far.

He is trying to find the sum of the interior angles.

Could you please complete Jacob's working to find the sum of the interior angles? Pause the video while you do this and press play when you're ready to work through the answer.

Okay, we can see that the sum of the angles in the triangles is 1,080 degrees, but we have those two extra vertices and the angles on those vertices add up to 180 degrees, so the sum of the extra angles is two lots of 180, which is 360 degrees.

So to get the sum of the interior angles of the hexagon, let's do a subtraction.

1,080, subtract 360 to get 720 degrees.

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

This task contains just one question, and here it is.

You've got three polygons, polygon A, polygon B, and polygon C, which have all been broken into triangles, but using different methods.

And underneath, you have a table where you need to fill in information about how polygon A, B, and C have been split up and then find the sum of the interior angles in those three polygons.

Pause the video while you have a go at this, and then press play when you're ready to work through the answers.

Okay, well done with that.

Let's now go through some answers.

In polygon A, we have 10 triangles.

The angles in those triangles sum to 1,800 degrees.

It has created one extra vertex in the middle of the polygon, and the angles around that vertex sum to 360 degrees.

Therefore, the sum of the interior angles in the polygon is 1,440 degrees.

In polygon B, we have four triangles.

They sum to 720 degrees, but we have that vertex in the middle, which has created three extra angles, which sum to 180 degrees, so the sum of the interior angles in the polygon is 540 degrees.

And with polygon C, we've got eight triangles which sum to 1,440 degrees, but we have two new vertices created.

One is inside the polygon where the angles add to 360, and one is on the bottom edge of the polygon where those two angles add to 180, so the sum of the extra angles is 540 degrees.

Therefore, the sum of the interior angles for the polygon is 900 degrees.

Excellent work with that.

So we can find the sum of the interior angles in a polygon by splitting it into shapes and can do it in lots of different ways, or we can also use sequences to find the sum of interior angles, and that's what we're gonna do now.

Sam is investigating sums of interior angles by manipulating one shape to create another.

So so far, he has a triangle which has three sides, and the interior angles sum to 180 degrees.

Sam says, "I can increase the number of sides by one "by adding an extra triangle," like so.

This now means he has a four-sided shape where the interior angles sum to 360 degrees.

And then if he adds an extra triangle like this, we have a five-sided shape where the interior angles sum to 540, and let's do it again.

We have a six-sided shape where the interior angles sum to 720.

Sam says, "Each time the number of sides increases by one, "the sum of the interior angles increases by 180 degrees." Can we think why that is? Well, that must be because we kept adding on extra triangles to increase the number of sides each time.

Therefore, every time we add another triangle, we add another 180 degrees.

Here we have Jun.

Jun is looking for patterns within the sums of interior angles.

Jun says, "The sums "of interior angles makes an arithmetic sequence," and we can see that because there is a constant difference between the sums of interior angles.

We start with 180, and then we keep adding 180 degrees on each time to get the sum of interior angles in the next polygon.

So if we have a sequence, can we generalise the sequence a bit more and maybe think about some algebra that represents a sequence? Jun says, "If we use the letter n to represent the number "of sides, then we could find the nth term of the sequence." So if n is the number of sides, when n is three, the sum of interior angles is 180.

When n is four, the sum of interior angles is 360.

Let's see if we can find the nth term now.

So Jun finds the nth term of this sequence.

However, he says, "You can't have a shape with one "or two sides," so his sequence starts at n equals three.

So that's why where he's written his values of n, he's lined it up so that the three is above 180 and the four is above 360, but there are blank spaces beneath the one and two, because you can't have a shape with one side or two sides.

Jun says, "The common difference is 180, "so let's write the sequence for 180n." Here it is.

And then Jun says, "The sequence "for 180n needs to be translated 360 "to make it into my sequence," because he's looking at, for example, where n is three.

His sequence of 180n gives 540, but he wants it to be 180.

There's a difference there of 360 between those numbers.

And the same as well when n is four, between 720 and 360, and when n is five, between 900 and 540.

So he wants to translate his sequence 360 to make it correct.

And that gives him this, 180n, subtract 360, where n is the number of sides.

Jun says, "This is the nth term for the sum "of the interior angles where n is the number of sides." Laura looks at this and considers if 180n minus 360 can be factorised.

She says, "The two terms have a common factor of 180," so we can take 180 out as a factor, and then what would go in the brackets? It would be n minus two.

So that means 180n subtract 360 is identical to 180 lots of n minus two.

And Laura says, "This is a factorised version "for the nth term of the sum of interior angles." And with this factorised version, we might be able to see how it relates to our methods of splitting a shape into triangles.

Laura says, "n is the number of sides that a polygon has.

"n minus two is the minimum number of triangles it can be split into," 'cause don't forget, the minimum number of triangles is always two less than the number of sides, and the 180 degrees is the sum of the interior angles of each triangle.

So what we have in brackets is a minimum number of triangles a polygon can be split into, and then we times it by 180 to get the sum of the interior angles.

So let's check what we've learned there so far.

The expression below represents the sum of interior angles in a polygon.

The expression is 180 times n minus two in brackets.

What does the n represent in this expression? Is it A, the number of sides in the polygon? Is it B, the minimum number of triangles that the polygon can be split into? Is it C, the sum of the interior angles in the polygon? Or is it D, the sum of the interior angles in each triangle? Pause a video while you make a choice, A, B, C, or D, then press play when you're ready for an answer.

The answer is A.

The n represents the number of sides in the polygon.

So if a polygon has 10 sides, n is 10.

If polygon has 12 sides, n is 12.

So in that case, what does n minus two represent? Same options again.

Pause the video while you choose A, B, C, or D and press play when you're ready for an answer.

The answer is B, n minus two represents the minimum number of triangles that the polygon can be split into, because the minimum number of triangles is always two less than the number of sides.

So what does the 180 represent? Pause the video while you make a choice, and press play when you're ready for an answer.

The 180 represents the sum of the interior angles in each triangle.

Here we have Andeep.

Andeep considers how the expression could be used to find the sum of interior angles in other polygons.

So Andeep says, "n is a number of sides in a polygon.

"So if a polygon has 12 sides, "then I could substitute 12 into the expression "to find the sum of its interior angles." So now he has 180 multiplied by, and in brackets, 12 subtract two.

This gives 180 multiplied by 10, where 10 is the minimum number of triangles it would've been split into if we'd split into triangles, and that gives 1,800.

So the sum of the interior angles is 1,800 degrees.

Let's do an example together, and then you go check what you've learned by doing a very similar example yourself.

Use the expression to find the sum of the interior angles in an icosagon, which is a 20-sided shape.

We can do that by substituting 20 in for n.

So we have 180 multiplied by, in brackets, 20 subtract two.

20 subtract two is 18, so we've got 180 multiplied by 18, which gives 3,240 degrees.

And now it's your turn.

Use the expression to find the sum of the interior angles in a triacontagon, that's a 30-sided shape.

Pause the video while you have a go at this, and then press play when you're ready for an answer.

Okay, we'll do that by substituting 30 for n, and then multiplying 180 by 28, which gives 5,040 degrees.

So what else could we do with this expression? Here we have Jacob.

Jacob says, "We have a relationship between the number "of sides of a polygon, n, "and the sum of its interior angles.

"So we could use it to find the number of sides "if we know the sum of the angles." So for example, a polygon has interior angles that sum to 3,240 degrees, how many sides does the polygon have? Perhaps pause the video at this point and think about how we might go about working this out, and then press play when you're ready to work through it together.

Well, one approach could be to make an equation.

On the left side of the equation, we have 3,240, that is the sum of the interior angles, and on the right-hand side of the equation, we have the expression that represents the sum of interior angles, where n is the number of sides.

That's 180 multiplied by n minus two.

So to find how many sides it has, we need to solve the equation.

We can do that by dividing both sides by 180 and then adding two to both sides to get n equals 20, so it must have 20 sides.

Here's another problem.

A regular polygon has interior angles that are each 162 degrees.

Now remember, in a regular polygon, all the angles are equal to each other.

They're all the same.

So how many sides does this polygon have? Well, we'll hear from Alex and Jacob in a moment, but maybe pause the video at this point and think about how you might approach this problem before pressing play to continue.

Well, Alex says, "If we know the sum of the interior angles, "we can find the number of sides, but we don't know the sum.

"We just know the size of each angle." Jacob says, "True, but we know it's a regular shape, "so we can write an expression for the sum "of the interior angles "which doesn't involve any additional variables, "so we can still solve this algebraically." Jacob says, "There are n sides "and n interior angles on this polygon.

"As it's regular, they are all 162 degrees.

"So the sum of interior angles can be written as 162n." Here we have 162n equals 180 multiplied by n minus two, where both sides of that equation represent the sum of the interior angles, but just in different ways.

We can then expand the brackets and rearrange the equation to get 18n equals 360, so n must be 20, therefore the polygon has 20 sides.

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

This task contains two questions and they're both displayed on the screen.

In question one, it says, "Find the sum "of the interior angles for each polygon." It gives you the name of the polygon plus the number of sides it has.

And in question two, it says, "A regular polygon's interior angles sum "to 21,420 degrees.

"How big is each of its interior angles?" Pause the video while you have a go at these and then press play when you're ready to work through the answers.

Okay, well done with that.

Here are our answers to question one.

Pause the video while you check these against yours, and then press play when you're ready for the answers for question two.

And then for question two, if the polygon has interior angles that sum to 21,420 degrees, we need to figure out how many sides it has.

So the polygon has 121 sides, and 121 interior angles.

Each angle in a regular polygon is equal, so each angle must be 177.

02 degrees to two decimal places.

Well done today, absolutely brilliant work.

Here's a summary of what we've learned in this lesson.

The focus of the lesson has been about exploring multiple different ways to find a sum of interior angles in a polygon, because there are lots of different ways we can do it, even more than the number of ways we've done in today's lesson.

Some of the ways involve splitting a shape into triangles and then using those triangles to find a sum of the interior angles.

And even with that method, there are lots of different ways we can split the shape into triangles.

But with some of those ways, it creates additional vertices inside the shape whose angles do not contribute to the interior angles of the overall shape.

That's okay, we've learned how to rectify that.

We subtract either 360 or 180 for each vertex that is made.

And then we've also seen how there's a numerical sequence that can be used to derive the sum of interior angles as well.

And because that sequence is arithmetic, it goes up in 180s, we can find the nth term for it, which creates a formula that can be used to find the sum of interior angles as well.

Great work today, well done.