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## Triangles and Polygons

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# Sum of interior angles of a polygon

## Video transcript

We already know that the
sum of the interior angles of a triangle add
up to 180 degrees. So if the measure
of this angle is a, the measure of this
angle over here is b, and the measure of
this angle is c, we know that a plus b plus
c is equal to 180 degrees. But what happens
when we have polygons with more than three sides? So let's try the
case where we have a four-sided polygon--
a quadrilateral. And I am going to
make it irregular just to show that whatever we
do here it probably applies to any quadrilateral
with four sides. Not just things that have right
angles, and parallel lines, and all the rest. Actually, that
looks a little bit too close to being parallel. So let me draw it like this. So the way you can think
about it with a four sided quadrilateral,
is well we already know about this-- the measures
of the interior angles of a triangle add up to 180. So maybe we can divide
this into two triangles. So from this point
right over here, if we draw a line
like this, we've divided it into two triangles. And so if the measure this
angle is a, measure of this is b, measure of
that is c, we know that a plus b plus c is
equal to 180 degrees. And then if we call this over
here x, this over here y, and that z, those are the
measures of those angles. We know that x plus y plus
z is equal to 180 degrees. And so if we want the
measure of the sum of all of the interior angles,
all of the interior angles are going to be
b plus z-- that's two of the interior
angles of this polygon-- plus this angle, which is
just going to be a plus x. a plus x is that whole angle. The whole angle for
the quadrilateral. Plus this whole angle, which
is going to be c plus y. And we already know a plus
b plus c is 180 degrees. And we know that z plus x plus
y is equal to 180 degrees. So plus 180 degrees, which
is equal to 360 degrees. So I think you see
the general idea here. We just have to figure out how
many triangles we can divide something into, and then we
just multiply by 180 degrees since each of those triangles
will have 180 degrees. Let's do one more
particular example. And then we'll try to do a
general version where we're just trying to figure out
how many triangles can we fit into that thing. So let me draw an
irregular pentagon. So one, two, three, four, five. So it looks like a little bit
of a sideways house there. Once again, we can draw
our triangles inside of this pentagon. So that would be
one triangle there. That would be another triangle. So I'm able to draw three
non-overlapping triangles that perfectly cover this pentagon. This is one triangle, the other
triangle, and the other one. And we know each of those
will have 180 degrees if we take the sum
of their angles. And we also know that the sum
of all of those interior angles are equal to the sum
of the interior angles of the polygon as a whole. And to see that, clearly,
this interior angle is one of the angles
of the polygon. This is as well. But when you take the sum
of this one and this one, then you're going to get
that whole interior angle of the polygon. And when you take the sum
of that one and that one, you get that entire one. And then when you take
the sum of that one plus that one plus that one, you
get that entire interior angle. So if you take the sum of all
of the interior angles of all of these triangles, you're
actually just finding the sum of all of the interior
angles of the polygon. So in this case, you have
one, two, three triangles. So three times 180
degrees is equal to what? 300 plus 240 is
equal to 540 degrees. Now let's generalize it. And to generalize
it, let's realize that just to get our
first two triangles, we have to use up four sides. We have to use up all the four
sides in this quadrilateral. We had to use up four
of the five sides-- right here-- in this pentagon. One, two, and then three, four. So four sides give
you two triangles. And it seems like, maybe,
every incremental side you have after that, you can
get another triangle out of it. Let's experiment with a hexagon. And I'm just going to try to
see how many triangles I get out of it. So one, two, three,
four, five, six sides. I get one triangle out
of these two sides. One, two sides of
the actual hexagon. I can get another
triangle out of these two sides of the actual hexagon. And it looks like I can
get another triangle out of each of the remaining sides. So one out of that one. And then one out of that
one, right over there. So in general, it
seems like-- let's say. So let's say that
I have s sides. s-sided polygon. And I'll just
assume-- we already saw the case for four sides,
five sides, or six sides. So we can assume that s
is greater than 4 sides. Let's say I have
an s-sided polygon, and I want to figure out how
many non-overlapping triangles will perfectly
cover that polygon. How many can I fit inside of it? And then I just have to
multiply the number of triangles times 180 degrees
to figure out what are the sum of the interior
angles of that polygon. So let's figure out
the number of triangles as a function of
the number of sides. So once again, four
of the sides are going to be used to
make two triangles. So those two sides
right over there. And then we have two
sides right over there. I can draw one triangle
over-- and I'm not even going to talk about
what happens on the rest of the sides of the polygon. You could imagine putting a
big black piece of construction paper. There might be other sides here. I'm not going to even
worry about them right now. So out of these two
sides I can draw one triangle, just like that. Out of these two sides, I can
draw another triangle right over there. So four sides used
for two triangles. And then, no matter how many
sides I have left over-- so I've already used four of
the sides, but after that, if I have all sorts
of craziness here. I could have all sorts
of craziness here. Let me draw it a little
bit neater than that. So I could have all sorts of
craziness right over here. It looks like every
other incremental side I can get another
triangle out of it. So that's one triangle out
of there, one triangle out of that side, one
triangle out of that side, one triangle out of that side,
and then one triangle out of this side. So for example, this
figure that I've drawn is a very irregular-- one, two,
three, four, five, six, seven, eight, nine, 10. Is that right? One, two, three, four, five,
six, seven, eight, nine, 10. It is a decagon. And in this decagon,
four of the sides were used for two triangles. So I got two triangles
out of four of the sides. And out of the other
six sides I was able to get a triangle each. These are six. This is one, two,
three, four, five. Actually, let me make sure I'm
counting the number of sides right. So I have one, two, three, four,
five, six, seven, eight, nine, 10. So let me make sure. Did I count-- am I just
not seeing something? Oh, I see. I actually didn't-- I have to
draw another line right over here. These are two different
sides, and so I have to draw another
line right over here. I can get another triangle
out of that right over there. And so there you have it. I have these two triangles
out of four sides. And out of the other
six remaining sides I get a triangle each. So plus six triangles. I got a total of
eight triangles. And so we can generally
think about it. The first four, sides we're
going to get two triangles. So let me write this down. So our number of triangles
is going to be equal to 2. And then, I've already
used four sides. So the remaining sides
I get a triangle each. So the remaining sides
are going to be s minus 4. So the number of triangles are
going to be 2 plus s minus 4. 2 plus s minus 4
is just s minus 2. So if I have an
s-sided polygon, I can get s minus 2 triangles that
perfectly cover that polygon and that don't overlap with
each other, which tells us that an s-sided polygon, if
it has s minus 2 triangles, that the interior angles
in it are going to be s minus 2 times 180 degrees. Which is a pretty cool result. So if someone told you that they
had a 102-sided polygon-- so s is equal to 102 sides. You can say, OK, the
number of interior angles are going to be 102 minus 2. So it's going to be 100
times 180 degrees, which is equal to 180 with two
more zeroes behind it. So it'd be 18,000 degrees
for the interior angles of a 102-sided polygon.