Video: Interior Angles of a Regular Polygon

Use the fact that the sum of measures of interior angles (in degrees) in an 𝑛-sided polygon is 180(𝑛 − 2) to calculate the measure of one interior angle in a regular polygon, or work backward from the interior angle to calculate the number of sides.


Video Transcript

In this video, we’re going to look at how to calculate the interior angle of a regular polygon.

First, let’s be clear what is meant by that word regular. Now, there are two diagrams on the screen, one of a regular hexagon and one of an irregular hexagon. Now, they’re both hexagons, so they both have six sides, but they look very different from each other. If you look carefully at the diagrams, you’ll see that the difference is this. In the regular hexagon, all of the sides are the same length and all of the interior angles are also all the same size. Whereas in the irregular hexagon, that isn’t the case.

So, in this video, we are focusing specifically on just those polygons that are regular. Now, just a reminder about interior angles, the interior angles of a polygon are those angles inside the shape itself, so those which I’ve marked in red in my diagram of a hexagon here. Now, as we’ve already seen, in the case of a regular polygon, all of those interior angles must be the same as each other. What we would like to know is how do we calculate the size of each of those interior angles in a regular polygon with a particular number of sides.

Now, we’ve seen previously that there is a formula for calculating the sum of the interior angles in any polygon. And it’s this formula here. So, the sum of the interior angles in an 𝑛-sided polygon is equal to 180 multiplied by 𝑛 minus two, where 𝑛 represents the number of sides. Now, notice that there’s no mention of the word regular here. So, this formula is true regardless of whether the polygon that we’re interested in is regular or irregular.

And just a brief reminder of where this formula came from, if you look at a polygon, and if you choose a particular corner such as this one here. If you then connect it to all the other corners of the polygon, so like I’ve done here, you will see that you divide the polygon up into triangles. And in this case, I’ve got four triangles. What you’ll notice, if you do this for a number of different polygons, is that the number of triangles you create is always two less than the number of sides.

So, here I had six sides, and indeed I created four triangles. Within each of those triangles, there are 180 degrees. And therefore, the total sum of the interior angles is the number of triangles multiplied by 180. And as there are always two less triangles than the number of sides, that’s where this factor of 𝑛 minus two comes from. So, that formula holds true for the sum of the interior angles regardless of whether the polygon you’re interested in is regular or irregular.

However, this video is specifically about regular polygons, and it’s about calculating the size of each individual interior angle rather than just the total sum. So, let’s think how we can use this formula to work out the size of each individual interior angle in a regular polygon. Well, if a polygon has 𝑛 sides, then it will also have 𝑛 interior angles. And all of them are the same because it’s a regular polygon. So, if we know the sum, the total of these interior angles. If we want to work out what each individual one is equal to, we just need to divide by the number of angles.

Which means we need to divide by 𝑛. Which means we can deduce this formula here. The interior angle in a regular 𝑛-sided polygon is equal to 180 times 𝑛 minus two divided by 𝑛, so the total sum divided by the number of interior angles that there are within the polygon. It’s really important to remember that this is only true if the polygon you’re looking at is regular. If it’s irregular, then all of the interior angles will be different sizes, so we can’t have a general formula to work them out.

So, now, let’s apply this formula to this question here which asks us to find the measure of one interior angle in the regular hexagon.

So, it’s just a question of using the formula that we have, but substituting the value of 𝑛. Now, 𝑛, remember, represents the number of sides. So, in the hexagon that’s six, so I’m just gonna substitute 𝑛 equals six into our formula for the interior angle. So, this tells me that each interior angle is 180 multiplied by six minus two, or four, and then I divide that by six. And if I work that out, it tells me that each interior angle is equal to 120 degrees.

So, you could answer this question for any regular polygon whatsoever as long as you know the number of sides. It’s just a question of substituting the correct value of 𝑛 into the formula that we’ve already worked out. Now, let’s look at a different type of question.

This question tells us that each interior angle of a regular polygon is 160 degrees. We’re then asked to calculate how many sides this polygon has.

So, this question is an example of working backwards. We’ve been given the size of each interior angle, and we want to work back to work out the number of sides. So, let’s think about how to approach this question. We know what each interior angle is, and we also know a formula for working out the interior angle. And if I remind you, it was this formula here, that the interior angle is 180 multiplied by 𝑛 minus two all over 𝑛, where 𝑛 represents the number of sides.

So, we can use these two pieces of information to form an equation. So, by setting these two things equal to each other, we have the following. 180 lots of 𝑛 minus two over 𝑛 is equal to 160. And that’s taking the formula for the size of the interior angle and the value that we know the interior angle is. So, now this problem has essentially become an algebra problem. We’ve got an equation, and we need to solve it in order to find out the value of 𝑛.

So, the first step I would take here. There’s an 𝑛 in the denominator of the left-hand side of the equation. So, in order to eliminate that from the denominator, I’m gonna multiply both sides of this equation by 𝑛. And when I do that, you’ll see I now have 180 lots of 𝑛 minus two is equal to 160𝑛. Next step, well, there are lots of different ways you could go through solving this equation. I’m gonna choose to expand the brackets on the left-hand side.

So, I have 180𝑛 minus 360 is equal to 160𝑛. Next, I’m gonna group all the 𝑛s on the left-hand side. So, I’m gonna subtract 160𝑛 from both sides of the equation, giving me 20𝑛 minus 360 is equal to zero. Then, I’m gonna add 360 to both sides, which gives 20𝑛 is equal to 360. Final step is just to divide both sides of the equation by 20. And that gives me 𝑛 is equal to 18, which is our answer for the number of sides that this polygon has.

So, just a reminder, this question involved working backwards. We knew the interior angle and we’re working back to work out the number of sides. Often when a question does involve working backwards, it’s a good idea to form an equation and then solve it algebraically to help you answer the problem. Now, it might be sensible just to check our answer. So, we can substitute the value of 18 back into the formula of the interior angle and check that we do indeed get 160 degrees. And you could do that yourself to verify that this is in fact the correct answer.

Okay, our final question, we’re given the diagram here. And the question says, is it possible to create this tessellating pattern for a regular octagon, a regular hexagon, and a square?

So, the question is if you were to create this pattern, would the shapes actually be regular? Now, let’s think out what this has got to do with interior angles. Well, what you notice is that, in part of this design, there is a particular point here where all three of those shapes come together. And where the interior angles of those three shapes are clustered together around a point. What this means is that the question is essentially saying, do the interior angles of these three shapes add up to 360 degrees? Because if they don’t, we’re either gonna have a gap between these shapes or we’re gonna have an overlap between them.

So, what we need to do is we need to think about the interior angles of each of these three shapes. So, a reminder then of the formula that we’re going to need, in an 𝑛- sided regular polygon, the interior angle is found in this way here. 180 multiplied by 𝑛 minus two over 𝑛. So, let’s work them all out. Regular octagon, first of all, well an octagon has eight sides. So, I’m just gonna be substituting 𝑛 equals eight into this formula. So, 180 multiplied by eight minus two over eight. And that gives 135 degrees for the interior angle of the octagon.

Now, the hexagon we’ve already seen in this video, but we can write it out again. So, 180 multiplied by six minus two over six. And as we saw before, that gave an interior angle of 120 degrees for the hexagon. The square, well you probably already know that the interior angle in a square is 90 degrees. You can verify that using the formula by substituting 𝑛 equals four, but we’ll just go with 90 degrees. So, there we have the three angles worked out.

So, I’ve labelled each of them on the diagram. And the question is then, well, do those three angles add up to 360? And of course, they don’t, the total of these three angles is, in fact, 345 degrees. Which means if you were trying to make this tessellating pattern out of regular shapes, you would in fact have a gap. So, the answer to the question, is it possible? No, it isn’t possible to do this.

So, to summarise then, in this video, we’ve looked at exactly what a regular polygon means. We’ve looked at the formula for calculating the interior angle in a regular polygon if you know the number of sides. We’ve looked at working backwards from knowing each interior angle to calculating the number of sides. And then, last, just a little bit of problem solving where this skill would be useful.

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