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## High school geometry

### Course: High school geometry > Unit 5

Lesson 3: Special right triangles# Area of a regular hexagon

CCSS.Math:

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

## Want to join the conversation?

- What is the formula of a hexagon?(17 votes)
- The formula to calculate the area of a regular hexagon with side length s:

(3 √3 s^2)/2

Remember, this only works for REGULAR hexagons. For irregular hexagons, you can break the parts up and find the sum of the areas, depending on the shape.

Hope that helped!(28 votes)

- instead of dividing the hexagon into 6 triangles wouldn't it be slightly easier to draw a hypothetical line from point f to point b and again from point e to point c turning it into 2 triangles and a rectangle?(7 votes)
- You can try it and see. The advantage to dividing the hexagon into six congruent triangles is that you only have to calculate the area of one shape (and then multiply that answer by 6) instead of needing to find the area of both a rectangle and a triangle.(10 votes)

- At7:04, isn't the area of an equilateral triangle (sqrt(3)*s^2)/4? I still get 3*sqrt(3), so I guess it's not as important as I thought...(7 votes)
- Well, you are actually right. That would be the special formula that gives you the area of equilateral triangles.

However the general area formula for triangles used in the video (A = 1/2*h*b), works for all triangles, including equilateral ones.

Using the special formula as suggested by you would have been quicker though, as you only need to know the side measurement of the equilateral, while the general formula requires the height and the base measurement.(7 votes)

- Couldn't you just divide it into separate triangles and add up the area of those?(2 votes)
- Yes, however formulas save time. This shape is small, but what about if it had 100 sides? Do you really want to calculate that many triangles. And each one of those triangles, you would need both the base and the height, which might not be given.(6 votes)

- What is the length of a side of a regular six sided polygon with radius of 8cm?(3 votes)
- Radius is the distance from the center to a corner. it is also important to know the apothem This works for any regular polygon.

Choose a side and form a triangle with the two radii that are at either corner of said side. You know both radii are 8 cm, which means you have an isosceles triangle.

You want to count how many of these triangles you can make. Basically each side will have one of these. this means each triangle will have an angle of measure 360/n, where n is the number of sides. In your case that is 360/6 =60. Since it is a scalene triangle you know the measure of the other two angles are the same. Also, you should know the angles of a triangle add up to 180. so in other words use some algebra to find the two other angles. Here that works out like this.

one angle is 60 and the other two are some other angle x where all three equal 180. So that works out to 60 + x + x = 180.

60 + 2x = 180

2x = 120

x = 60.

So this shows al four angles are 60 degrees, which means not only is it a scalene triangle, but an equilateral triangle. This means all sides are the same. And since we know the radii that means the remaining side is the sme measure at 8 cm.

If these were not equilateral you would have to use the apothem and the Pythagorean theorem.(3 votes)

- i dont get this. whats going on?(2 votes)
- Sal is explaining how to get the area of a regular hexagon. Try rewatching the video if you don't understand.(4 votes)

- what about a polygon? I'm confused.(1 vote)
- A hexagon is a polygon as are squares, triangles, rectangles, octagons and many other shapes. For each shape the formula to find the area will be different.(4 votes)

- At0:18you failed to mention that all exterior angles are congruent and have the same measure as well as the interior angles. Of course, even if the hexagon isn't regular and all sides aren't congruent, the exterior angles could still be congruent provided they are attached the right kind of polygon. Anyways, I just felt like pointing that out because it really itched my brain. Side note: Thanks for the great math videos, they really help!(1 vote)
- I feel like defending Khan here, and I don't want to be a jerk, but:

He doesn't need to point out that the exterior angles are congruent because it's not relevant to what he's trying to solve: the area of the hexagon. Why mention it if it could be confusing the audience of why it's important?

He also told us that the angles all have the same measure at0:18, which also means the interior angles are congruent, as by the Definition of Congruent Angles.

Your second argument was confusing, yet I get what you mean. Imagine that AB and DE were 4 units long, which would keep the interior angles at 120 degrees and thus the exterior angles congruent. Yet, again, the argument is about exterior angles, and exterior angles are not needed to find the area.(4 votes)

- OK, so each triangle has 180°. 6x180=1080°, not 360°. I don't see why this doesn't work out.(2 votes)
- The total degrees of a triangle is 180 degrees, but in the video the 360 degrees is the total of all the top angles AGB, BGC, CGD, etc. So it is really 60 degrees times 6 which = 360. Hope this helps.(2 votes)

- can't you just use ((sqrt(3)s^2)/4) multiplied by six since the first part is the formula to find the area of equilateral triangles, and then since there are 6 equilateral triangles in a regular hexagon, you can multiply it by 6?(2 votes)
- Yes your method works. When you multiply the formula for an equilateral triangle by 6, you get the formula for the area of a regular hexagon.(2 votes)

## 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.