Special right triangles
Area of a Regular Hexagon Using what we know about triangles to find the area of a regular hexagon
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- We're told that A B C D E F 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 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 its two square roots of three
- So this side right over here is two square roots of three
- This side right over here is two square roots of three
- And I could just go around hexagon,
- every one of their sides is two square roots of three
- They want us to find the area of this hexagon
- Find the area of A B C D E F
- And the best way to find the area especially of regular polygons,
- Is to try to spilt up into triangles
- And hexagons are a bit of special case
- Maybe in the future videos,
- we'll think about the more general case of any polygon
- With the hexagon, what you can think about is if we
- If we take this point right over here and let's call this point G
- Now 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 coz this isn't a circle [1:08 7]
- But we could say it's equidistant from all of the vertices
- So 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 I'm gonna draw some of those that I just talked about
- So that is GE
- There's GD
- There's GC
- All of these lines 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 the 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 if 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
- Coz G is in the center and they all have this third common side
- Of two square roots of three
- So all them by side-side-side they are all
- They're 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
- And maybe we'll call that X
- That's angle X, that's X, that's X, that's x, that's x;
- And if you'd add them all up we've gone around the circle
- We've gone 360 degrees and we have six of these Xs
- So you get six X is equal to 360 degrees
- you divide both sides by six, you get X is equal to
- X is equal to 60 degrees
- 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 a trivial pursuit piece
- But we know that they're definitely isosceles triangle
- 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 are two base angles
- This angle is going to be congruent to that angle
- And we could call that Y right over there
- So you have Y+Y which is 2Y+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've 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 is also two square roots of three
- And this is also two square roots of three
- So pretty much all of these green lines are 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 that information
- We can use that information to figure out;
- Actually we don't really 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 it by six
- So let's focus on let me focus on this triangle right over here
- 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 have 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 wanna find the area of this broader
- 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 it 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
- DH is going to be the square root of three
- And we are or hopefully we already recognized
- 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 it
- This is two square root of three or we don't really need it
- What we really need to figure out is this altitude height
- And from 30-60 degree 90 30-60-90 triangles
- we know that side opposite the 60 degree side is square root of three
- Is a square root of three
- Times the side opposite the 30 degree side
- So this is going to be square root of three
- Times the square root of three,
- times the square root of three
- Square root of three times square root of three
- Is obviously just 3
- So this altitude right over here is just going to be three
- So if we want the area of this triangle right over here,
- Which is this triangle right over here
- Is just one half base times height
- So the area of this little sub slice
- is just one half 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 let me rewind this a little bit
- Coz now we have the base and the height of the whole thing
- If we care about, if we care about the area triangle GDC
- So now I'm looking at
- Now I'm looking at this entire triangle right over here
- This is equal to one half times base times height
- Which is equal to one half
- 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
- So it's this whole thing right over here
- So times two square roots of three
- And then we wanna multiply that times our height
- And that's what we just figure out using 30-60-90 triangles
- Our height is three
- So times three One half and two cancel out
- We're left with three square roots of three
- That's just the area of one of these little wedges right over here
- If we wanna find the area of the entire hexagon
- We just have to multiply that by six
- Coz there are six of these triangles there
- So this is going to be equal to six times three square roots of three
- Which is 18 square roots of three and we're done
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At 5:31, how is the moon large enough to block the sun? Isn't the sun way larger?
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