Video Transcript
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 20-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 180 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 180
degrees, which means that, for the quadrilateral, the total sum of all its angles
will be two lots of 180 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 180 degrees. Therefore, as I’ve got four of
them, the sum of the interior angles in the hexagon will be four lots of 180
degrees. So the sum of the interior angles
for the hexagon is 720.
Now you could try this method
yourself perhaps with some other polygons, 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 180
degrees in it, which means the sum of the interior angles will be 180 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 180. 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 180 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 180 multiplied by three, which tells us that the sum of the interior
angles is 540 degrees. And we could answer the same
question for any polygon, a 10-sided shape, a 12-sided, a 38-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 1620 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
1620. So I can write down the equation
180 lots of 𝑛 minus two is equal to 1620. 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 algebraic problem. I want to solve this equation, so
the first step is to divide both sides of this equation by 180. 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 11-sided polygon that
has a sum of interior angles equal to 1620.
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
halve 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 180 times three. That’s five minus two because there
were five sides, which gives us 540 degrees for the total sum. So this means that all of these
angles added together, so 121 plus 110 plus 85 plus two lots of 𝑦 must be equal to
540. 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 121, 110, and 85 together, so now I have the equation two 𝑦 plus 316 is equal
to 540.
The next step to solving this would
be to subtract 316 from both sides of the equation, so now I have two 𝑦 is equal to
224. 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 112 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.
