Special right triangles
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.