Lesson video

Hello, my name is Mr Clasper and today we are going to be learning, how to solve problems involving exterior angles.

Before we begin today's lesson, I would recommend that you look at our lesson on finding missing exterior angles.

If you've already covered that lesson, you should be fine if not have a go and then move on to this lesson.

In his diagram, we have an equilateral triangle which has three 60 degree angles and three 120 degree angles, And from previous work we should remember that the sum of exterior angles on any polygon is 360 degrees.

Now one way to calculate this Would be to take 360 degrees and divide it by three, as we have three exterior angles, and this will give us our answer of 120 degrees if we wanted to find the exterior angle.

We can use a general formula to help us solve these problems in a more efficient manner.

So if we say that e is our exterior angle and n is the number of sides on the polygon, then that means the exterior angle must be equal to 360 divided by the number of sides or e is equal to 360 all over n.

Lets have a look at an example, this is a regular hexagon am going to need to calculate the size of the exterior angle a.

Now to do this, we can use our formula.

So we need to calculate 360 divided by the number of sides.

Well the shape has six sides and six exterior angles, therefore am going to divide my 360 degrees or my exterior angle sum, into six equal parts.

So 360 divided by 6 and this gives us a value of 60.

Therefore the value of a, must be 60 degrees.

We can also use this formula to find the number of sides a shape has, if we know the size of the exterior angle.

So starting with our original formula, if we multiply both sides of this formula by n that means that n multiplied by e must be equal to 360 and this makes sense.

If we think about our last example our hexagon had six sides, and each of its six exterior angles was 60 degrees, and 6 multiplied by 60 would give us 360 degrees.

Now if I divide both sides of this equation now by e, I get a new formula, a re-arranged formula.

Which is n is equal to 360 divided by e or the number of sides is equal to 360 degrees divided by the size of the exterior angle.

Lets see if we can apply our new formula.

The diagram shows part of a polygon, we know that this polygon has an exterior angle of 20 degrees, but we need to know how many sides this polygon has.

So using our new re-arranged formula, we know that the number of sides must be equal to 360, divided by the size of the exterior angle.

So if I calculate 360 divided by 20 which is our exterior angle, this gives us a value of 18.

Therefore, the number of sides this polygon must have would be 18.

Here are some problems for you to try.

Pause the video to complete your task, click resume once you have finished.

And here are your solutions to question one.

So again, remember to find the number of sides on a polygon, we can calculate 360 degrees divided by the size of the exterior angle, and this will give us a solution to each of our problems. Lets take a look at this example, so again, we've been given a part of a polygon and we need to work out how many sides this polygon has.

Now this question is slightly different because we have been given, the interior angle of this polygon.

We are told that the interior angle is 150 degrees, what we need is the exterior angle in order to use one of our formulas.

But we know, that the exterior angle and the interior angle must have a sum of 180 degrees, as they appear on a straight line.

That means that the exterior angle of our polygon, must be 30 degrees.

Now I know this, I can use my second formula again.

So if I calculate 360 divided by 30, this will tell me the number of sides this shape must have.

And if we calculate this we are told that n must be equal to 12, therefore the polygon must have 12 sides.

Here is a question for you to try.

Pause the video to complete your task and click resume once you have finished.

And here is your solution.

So unfortunately, David has calculated with his interior angle and he needs to make sure he uses his exterior angle for the formula that we have been using.

So the exterior angle on this polygon would actually be 72 degrees, meaning that we needed to calculate 360 divided by 72 which gives us an answer of five.

That means that our polygon has five sides or in other words is a regular pentagon.

Here are some questions for you to try.

Pause the video to complete your task and click resume once you have finished.

And here are your solutions.

So again, make sure you don't make the same mistake that David did in the previous example, you need to calculate your exterior angle first.

Once you have your exterior angle, if you divide 360 by your exterior angle you will be able to calculate the number of sides given.

Here is your last question, this one is a tough one be careful.

Pause the video to complete your task and click resume once you have finished.

And here is the solution to your final problem.

This one is quite tricky, so, if I know that angle BCE is equal to 105 degrees, then I also know, that the exterior angle of my smaller polygon plus the exterior angle of my larger polygon, must also have a sum of 105 degrees.

From this point its a case of investigating what we know about certain polygons.

And all we need to do is to find a polygon whose exterior angle or one of their exterior angles, could be added to the exterior angle of another polygon to give us 105.

And if we think about this when I have a look at a hexagon, a hexagon has six sides and its exterior angles are all 60 degrees, and an octagon which has eight sides has exterior angles which are all 45 degrees and if we add those two exterior angles together our 60 degree exterior angle for a hexagon and 45 degree angle for an exterior angle of an octagon the sum of those would be 105.

Therefore, the smaller polygon has six sides and the larger polygon has eight sides.

And that brings us to the end of our lesson.

So now, we've been able to solve problems involving exterior angles.

Why not give the exit quiz a go and boost your confidence.

I'll hopefully see you soon.