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CCSS.Math:

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 of the sides they say it's two square roots of three so this side right over here is two square roots of three this side over here is two squared so three and I can just go around the hexagon every one of their sides is two square roots three 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 let's 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 can think about is if we take 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 the 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 the GD is the same thing as G C is the same thing as G B which the same thing as G a which is the same thing as G F 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 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 add if we go all the way around the circle if we go all the way around a circle like that we've gone 360 degrees and we know that these triangles 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 two square roots of three so all of them by side-side-side they are all 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 is going to be the same for all six of these all six of these triangles over here maybe I 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 six you get X is equal to X is equal to sixty degrees X is equal to sixty degrees all of these are equal to sixty degrees now there's something interesting we know that these triangles for example triangle GBC and we can do that for any of these six triangles it looks kind of like a Trivial Pursuit piece that 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 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 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 and we're dealing with an equilateral triangle which means that all the sides have the same length so this is two square roots of three then so is this this is also two square roots of three and this is also two square roots of three so pretty much all of these green lines or two square roots of three and we already knew because it's a regular hexagon that the every out each side of the hexagon itself is also two square roots of three so now we can essentially use the that information we can 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 we can just multiply by six so let's focus on let me focus on this triangle right over here and think about how we can find its area we know that length of DC is two square roots of three we can drop an altitude over here we can drop an altitude just like that and then we if we drop an altitude we know that this is 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 you just 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 broader truck of this little slice of the pie right over here we can just find the area of this slice or the sub slice and then multiply by two 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 D H is going to be the square root of three and we we hopefully we've already recognizes 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 three we know and we already actually did calculate this is two square roots of three although we don't really need it what we're really needing to figure out is this altitude height and from 30 60 degree nine 30 60 90 triangles we know that the side opposite the 60 degree side square root as 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 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 be let's 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 let me rewind this a little bit because now we have the base of the height of the whole thing if we care about if we care about the area of triangle G DC so now I'm looking at 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 two square roots of three is 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 to 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 6 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