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# Area of a regular hexagon

Using what we know about triangles to find the area of a regular hexagon. Created by Sal Khan.

Video transcript

We're told that ABCDEF
is a regular hexagon. And this regular part--
hexagon obviously tells us that we're
dealing with six sides. And you could just count that. You didn't have to be
told it's a hexagon. But the regular
part lets us know that all of the sides, all six
sides, have the same length and all of the interior
angles have the same measure. Fair enough. And then they give us the
length of one of the sides. And since this is
a regular hexagon, they're actually giving us
the length of all the sides. They say it's 2
square roots of 3. So this side right over
here is 2 square roots of 3. This side over here is
2 square roots of 3. And I could just go
around the hexagon. Every one of their sides
is 2 square roots of 3. They want us to find the
area of this hexagon. Find the area of ABCDEF. And the best way
to find the area, especially of
regular polygons, is try to split it
up into triangles. And hexagons are a
bit of a special case. Maybe in future
videos, we'll think about the more general
case of any polygon. But with a hexagon, what
you could think about is if we take this
point right over here. And let's call this
point G. And let's say it's the center of the hexagon. And when I'm talking about
a center of a hexagon, I'm talking about a point. It can't be equidistant
from everything over here, because
this isn't a circle. But we could say
it's equidistant from all of the vertices,
so that GD is the same thing as GC is the same thing as GB,
which is the same thing as GA, which is the same thing as GF,
which is the same thing as GE. So let me draw some of those
that I just talked about. So that is GE. There's GD. There's GC. All of these lengths are
going to be the same. So there's a point
G which we can call the center of this polygon. And we know that this length
is equal to that length, which is equal to that length, which
is equal to that length, which is equal to that length,
which is equal to that length. We also know that
if we go all the way around the circle like that,
we've gone 360 degrees. And we know that these
triangles are all going to be congruent
to each other. And there's multiple ways
that we could show it. But the easiest way is,
look, they have two sides. All of them have this
side and this side be congruent to each other
because G is in the center. And they all have
this third common side of 2 square roots of 3. So all of them,
by side-side-side, they are all congruent. What that tells us is,
if they're all congruent, then this angle, this interior
angle right over here, is going to be the
same for all six of these triangles over here. And let me call that x. That's angle x. That's x. That's x. That's x. That's x. And if you add them all up,
we've gone around the circle. We've gone 360 degrees. And we have six of these x's. So you get 6x is
equal to 360 degrees. You divide both sides by 6. You get x is equal
to 60 degrees. All of these are
equal to 60 degrees. Now there's something
interesting. We know that these triangles--
for example, triangle GBC-- and we could do that for
any of these six triangles. It looks kind of like a
Trivial Pursuit piece. We know that they're
definitely isosceles triangles, that this distance is
equal to this distance. So we can use that
information to figure out what the other angles are. Because these two base angles--
it's an isosceles triangle. The two legs are the same. So our two base
angles, this angle is going to be
congruent to that angle. If we could call that
y right over there. So you have y plus
y, which is 2y, plus 60 degrees is going
to be equal to 180. Because the interior
angles of any triangle-- they add up to 180. And so subtract 60
from both sides. You get 2y is equal to 120. Divide both sides by 2. You get y is equal
to 60 degrees. Now, this is interesting. I could have done this with
any of these triangles. All of these triangles
are 60-60-60 triangles, which tells us-- and
we've proven this earlier on when we first
started studying equilateral triangles--
we know that all of the angles of a
triangle are 60 degrees, then we're dealing with an
equilateral triangle, which means that all the sides
have the same length. So if this is 2 square
roots of 3, then so is this. This is also 2
square roots of 3. And this is also 2
square roots of 3. So pretty much all
of these green lines are 2 square roots of 3. And we already knew, because
it's a regular hexagon, that each side of
the hexagon itself is also 2 square roots of 3. So now we can essentially
use that information to figure out--
actually, we don't even have to figure this part out. I'll show you in a
second-- to figure out the area of any one
of these triangles. And then we can
just multiply by 6. So let's focus on this
triangle right over here and think about how
we can find its area. We know that length of DC
is 2 square roots of 3. We can drop an
altitude over here. We can drop an altitude
just like that. And then if we drop
an altitude, we know that this is an
equilateral triangle. And we can show very
easily that these two triangles are symmetric. These are both 90-degree angles. We know that these two are
60-degree angles already. And then if you look
at each of these two independent triangles, you'd
have to just say, well, they have to add up to 180. So this has to be 30 degrees. This has to be 30 degrees. All the angles are the same. They also share
a side in common. So these two are
congruent triangles. So if we want to find the area
of this little slice of the pie right over here,
we can just find the area of this slice,
or this sub-slice, and then multiply by 2. Or we could just find this
area and multiply by 12 for the entire hexagon. So how do we figure out
the area of this thing? Well, this is going to be
half of this base length, so this length right over here. Let me call this
point H. DH is going to be the square root of 3. And hopefully we've
already recognized that this is a
30-60-90 triangle. Let me draw it over here. So this is a 30-60-90 triangle. We know that this length over
here is square root of 3. And we already
actually did calculate that this is 2
square roots of 3. Although we don't
really need it. What we really need to figure
out is this altitude height. And from 30-60-90
triangles, we know that the side opposite
the 60-degree side is the square root
of 3 times the side opposite the 30-degree side. So this is going
to be square root of 3 times the square root of 3. Square root of 3 times
the square root of 3 is obviously just 3. So this altitude right over
here is just going to be 3. So if we want the area
of this triangle right over here, which is this
triangle right over here, it's just 1/2 base times height. So the area of this
little sub-slice is just 1/2 times our base,
just the base over here. Actually, let's
take a step back. We don't even have to
worry about this thing. Let's just go straight to
the larger triangle, GDC. So let me rewind
this a little bit. Because now we have the base and
the height of the whole thing. If we care about the area
of triangle GDC-- so now I'm looking at this entire
triangle right over here. This is equal to 1/2
times base times height, which is equal to
1/2-- what's our base? Our base we already know. It's one of the
sides of our hexagon. It's 2 square roots of 3. It's this whole thing
right over here. So times 2 square roots of 3. And then we want to multiply
that times our height. And that's what we just figured
out using 30-60-90 triangles. Our height is 3. So times 3. 1/2 and 2 cancel out. We're left with 3
square roots of 3. That's just the area of
one of these little wedges right over here. If we want to find the
area of the entire hexagon, we just have to
multiply that by 6, because there are six of
these triangles there. So this is going to be equal
to 6 times 3 square roots of 3, which is 18 square roots of 3. And we're done.