In this video, we are going to look at how to calculate the sum of the interior angles of a polygon. So there are a couple of words in that title that we need to make sure we’re familiar with, and the first of those is a polygon.
A polygon is just any straight-sided 2D shape. So there are a number of examples drawn on the screen here. And you need to be familiar with the names of polygons with different numbers of sides. So I have drawn here polygons with three, four, five, six, seven, and eight sides and we’ll just review the names for those ones as a starting point. So the names of each of these polygons are on the screen. A three-sided shape of course is a triangle. A general four-sided polygon is a quadrilateral. And of course, there are lots of different types of quadrilaterals that you’ll be familiar with, squares, and rectangles, and parallelograms, and so on. A five-sided polygon is called a pentagon. A six-sided polygon is a hexagon, seven-sided polygon is a heptagon, sometimes occasionally referred to as a septagon, and an eight-sided polygon is an octagon. Now there are names for polygons with many more sides than this. For example, a twenty-sided polygon is called an icosagon. But these are some of the main ones that you need to be familiar with.
The other word in that title that we need to understand is the word interior. When we’re talking about the interior angles in a polygon, now these are the angles inside the shape. So here, they are labelled in red within each shape. It’s worth noting of course that there are the same number of interior angles as there are sides to the polygon. So the hexagon has six sides and it has six interior angles. Now the polygons I have drawn are what’s referred to as irregular polygons, which means that their sides are not all the same length, and also the interior angles are not all the same. If they were the same, they would be referred to as regular polygons.
We’re going to look specifically at what the sum of the interior angles is, in each of these polygons. Okay. So we’re going to look at the sum of interior angles in a quadrilateral and then a hexagon specifically, to start off with. Now this method relies on the fact that in a triangle there are a hundred and eighty degrees, which is a fact that you would’ve seen and perhaps proved before. What I’m gonna do in the quadrilateral is, I’m gonna pick one of the vertices, and I’m gonna pick this one here, and then I’m gonna connect that vertex, or corner, to all the other vertices to which it isn’t already connected. So in the quadrilateral, that’s only the one directly opposite it. What you see now is, I’ve divided that quadrilateral up into two triangles. Now if I just draw on the angles, each of these triangles, well the sum of the angles in each triangle, will be a hundred and eighty degrees, which means that for the quadrilateral, the total sum of all its angles will be two lots of a hundred and eighty degrees because we have two triangles. So for a quadrilateral, the sum of its interior angles is three hundred and sixty degrees, and you may already be familiar with that fact.
Now let’s do the same thing for the hexagon. So I’m gonna pick one of the vertices and again, I’m gonna join it to all of the other vertices to which it isn’t already joined. So I’ve picked this top vertex here and what you can see by making those joins is, I now have one, two, three, four triangles within this hexagon. Again, if I draw on all the interior angles here, you’ll see that the sum of the interior angles in the hexagon is the same as the sum of the interior angles in these four triangles together. Now as we’ve already said, the sum of the angles in each triangle be a hundred and eighty degrees. Therefore, as I’ve got four of them, the sum of the interior angles in the hexagon will be four lots of a hundred and eighty degrees. So the sum of the interior angles for the hexagon is seven hundred and twenty degrees.
Now you could try this method yourself perhaps with some other polygons. So perhaps a pentagon, or an octagon, or whatever you like really, and see how many triangles you create using this method each time. What we’d like to do is come up with a general rule that we’d be able to use for any polygon with any number of sides. So let’s look at these two and perhaps ones you’ve done yourself. What you’ll notice for the quadrilateral which has four sides, we were able to create two triangles, and for the hexagon which has six sides we were able to create four triangles. So the number of triangles is always two less than the number of sides. So let’s generalize for what we call an 𝑛-sided polygon where 𝑛 just represents the number of sides. Well as we said, the number of triangles will be two less than the number of sides. So if the number of sides is 𝑛, then the number of triangles will be 𝑛 minus two. And each of those triangles has a hundred and eighty degrees in it, which means the sum of the interior angles will be a hundred and eighty multiplied by 𝑛 minus two. And so this gives us a general formula that we can use to work out the sum of the interior angles in any 𝑛-sided polygon. I just need to subtract two from the number of sides and multiply it by a hundred and eighty.
So let’s look at answering a couple of questions on this. The first question asks us to find the sum of the interior angles in a pentagon. So remember, our formula for calculating the sum of the interior angles was a hundred and eighty multiplied by 𝑛 minus two, where 𝑛 represents the number of sides. A pentagon, remember, has five sides, so we’re just gonna be substituting 𝑛 equals five into this formula. So we have the sum of the interior angles in a pentagon is a hundred and eighty multiplied by five minus two, which of course is a hundred and eighty multiplied by three, which tells us that the sum of the interior angles is five hundred and forty degrees. And we could answer the same question for any polygon, a ten-sided shape, a twelve-sided, a thirty-eight-sided shape. We could answer in exactly the same way just by substituting the value of 𝑛 into our formula.
Okay. Let’s look at a different question. This question says the sum of the interior angles in a polygon is one thousand six hundred and twenty degrees. How many sides does this polygon have?
So in this question we’re essentially being asked to work backwards. We’re given the sum of the interior angles and we’re going back to work out the number of sides. So remember, here’s our formula for the sum of the interior angles and we can use this to form an equation. We know that this sum is equal to one thousand six hundred and twenty. So I can write down the equation a hundred and eighty lots of 𝑛 minus two is equal to one thousand six hundred and twenty. Now being asked to find the number of sides, so essentially what that’s saying is solve this equation to work out the value of 𝑛, because remember 𝑛 represents the number of sides. So now this has essentially become an algebra problem. I want to solve this equation, so the first step is to divide both sides of this equation by a hundred and eighty. Once I’ve done that, that gives me the line of working out 𝑛 minus two is equal to nine. The next step then is just to add two to both sides of this equation, and this will give me the answer to the problem, 𝑛 is equal to eleven. So it’s an eleven-sided polygon that has a sum of interior angles equal to one thousand six hundred and twenty.
Okay, a final question then. We’ve been given a diagram and we’re asked to calculate the measure of angle 𝐴𝐵𝐶. So that’s this angle here, when I go from 𝐴 to 𝐵 and then to 𝐶. The angle is currently labelled as 𝑦. So we’re going to need our formula for the sum of the interior angles at some stage, so I’ve written that down again here. And let’s look at what we’ve got. If we count the number of sides in the diagram, we have five sides, so 𝑛 is equal to five. So our strategy here, we can work out the total sum of the interior angles by using this formula. We know three of them, so we’ll be able to subtract that. And then the remaining two angles are both labelled as 𝑦, which means they’re the same. So whatever is left over, if we have it, that will give us the measure of angle 𝐴𝐵𝐶.
So let’s go through the working for this. So the sum of the interior angles is a hundred and eighty times three. That’s five minus two because there are five sides, which gives us five hundred and forty degrees for the total sum. So this means that all of these angles added together, so a hundred and twenty-one plus a hundred and ten plus eighty-five plus two lots of 𝑦, must be equal to five hundred and forty. Now we could write that down as an equation. And now we have this equation, we just need to go through the steps to solve it in order to work out the value of 𝑦. So first of all, if I just combine those a hundred and twenty-one, a hundred and ten, and eighty-five together, so now I have the equation two 𝑦 plus three hundred and sixteen is equal to five hundred and forty. The next step to solving this would be to subtract three hundred and sixteen from both sides of the equation, so now I have two 𝑦 is equal to two hundred and twenty-four. And then to find 𝑦, I can halve both sides of the equation. And that gives me a final answer of 𝑦 is equal to a hundred and twelve degrees.
So to summarise, in this video we’ve looked at the names of some common polygons. We’ve looked at how to calculate the sum of their interior angles, based on the number of sides. We’ve seen how to work backwards from knowing the sum of the interior angles to calculate the number of sides, and then looked at a problem associated with this.