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we now know how to find the area of an equilateral triangle what I want to do in this video is attempt to find the area of a and I know I'm mispronouncing in a Koch or coach snowflake and the way you construct one is you start with an equilateral triangle and then on each of the sides you split them into thirds and then the middle third you put another smaller equilateral triangle and that's after one pass and on the next pass you do that for all of the sides here so a little one here here here here here here here I think you get the general idea that's the next pass on the next pass you do it for all of these sides and what's really neat about this and we showed this in a previous video is that you have a figure here that has an infinite perimeter but we're about to see in this video it actually has a finite area which is kind of interesting to think about so let's start with a clean a clean equilateral triangle right over here we're gonna assume that each of the sides have length s so it's going to be a clean equilateral triangle each of the sides dry that a little bit neater each of the sides have length s and so what I'm going to do is I'm going to keep track of two things I'm going to keep track of the sides in this triangle as we as it turns into a snowflake I'm gonna keep track of the number of sides and I'm gonna keep track of the area after each pass of adding more smaller triangles so this is going to be our count of the area actually let me give myself a little bit more real estate just because I'm a feeling I might need to use it so I'm going to keep this is the sides I'm gonna write up here and then this is our running count of the area down here so right when we start we have three sides we have three sides and our area we already figured out in a previous video is going to be if we assume each of the sides or length s is going to be square root of 3 square root of 3 s squared over 4 fair enough that was just a simple equilateral triangle now we're going to take each of these sides divide them into thirds so we're going to take each side divide it into thirds and then in that middle third we're going to add another smaller equilateral triangle so it'll look like that on that side right over there and I want to I want to think about what we're doing to each side right here so before I did this this was just one side then I split it into thirds and that middle third I put I essentially put two sides in there I put an equilateral triangle so one side has now turned into one two three four sides so every time we do a pass of the making the snowflake more intricate each side will turn into four sides so you can imagine if we do this on all three sides we have four times three which is now twelve sides so if you multiply this times 4 so times for this gets us to twelve sides now and we can count them out just to make sure where our logic is correct 1 2 3 4 5 6 7 8 9 10 11 12 sides and now what is the area now what's going to be the area of our original yellow equilateral triangle plus the area of each of these smaller ones and what's the area of each of these smaller ones well first of all we have three of them there are three of these each of each of these smaller ones and then we use the formula for the area of an equilateral triangle again so it's going to be square root of 3 times s squared but now the length of each of the sides of each of these equal these smaller equilateral triangles they aren't s anymore they are s over 3 remember this this length right over here is s over 3 so this is going to be s over 3 as well every pass the sides of the equilateral triangle become one-third of the previous pass so this is it's not going to be s squared anymore it's going to be s over 3 squared and then all of that all of that over 4 then let's do another pass so we're gonna add these triangles right over here I'm gonna add these right over there and this is the last pass relax tree attempt to draw all of the triangles over there so how many how many sides am I going to have first of all after I do another pass well the previous pass I had 12 sides each of those 12 sides are now going to turn into 4 new sides when I had these little orange bumps there so I'm gonna multiply it times 4 again I'm gonna multiply it times 4 so now I'm going to have 48 sides I'm gonna have 48 sides and how many new triangles so what's the area well it's going to be the yellow area plus the blue area plus the orange area so how many new orange triangles do I have well I'm adding a new orange triangle to each of the sides for the previous pass in the previous past I had 12 sides so now I'm going to add 12 orange triangles I'm going to add 12 orange triangles and actually let me write that I'll just write 12 orange triangles but it's really I just multiplied it times four and then I'm going to have times the square root of three and now this isn't going to be SS over three anymore these are going to be s over nine these have one-third of the dimensions of these blue triangles so this is going to be s over nine s over nine squared s over nine squared over four and so I think you might start to see the pattern building if we do another pass after this one move to the right a little bit what will that look like let me do this in a different color that I haven't used yet let me see I haven't used this pink yet so now I'm going to have I'm going to have the previous number of sides that's my number of new triangles 48 times the square root of three times s over I'm going to now these are gonna be even one third of this s over 27 s over 27 to the second power all of that over four and I'm going to keep adding an infinite number of terms of this to get the area of a true Coach snowflake so I'm just gonna keep doing this over and over again so the trick really is is finding the this infinite sum and seeing if it actually if we get a finite number over here so the first thing I want to do just to simplify it let me just rewrite it a little bit let me rewrite it a little bit differently over here so the first thing that's kind of that might be obvious is that we can factor out a square root of 3 s squared over 4 so let me factor that out so if we factor out a square root of 3 s squared over 4 from all of the terms then this term right over here will become a 1 this term right over here is going to become a 3 let's see we factored out a square root of 3 we factor out a 4 and we factored out the S squared we factor out only the S squared so now it's going to have plus 3 x times 1/3 3 times third squared that's all we have left here we have the 1/3 squared and then we have this 3 and I'm not simplifying this on purpose so that we see a pattern emerge and then this next term right over here plus so this 12 is still going to be there but I'm going to write that as 3 times 4 3 times 4 and then we're gonna lose it this is we're factoring out the square root of 3 we're factoring out the 4 we're factoring out the S squared and so we're gonna be left with 3 squared that's what this is down here squared so this is 1 over 1/3 squared 1/3 squared and then that squared so that's what we're left with with that orange term and then we go to the pink term this pink term this is 48 is just 3 times 4 times 4 3 times 4 I'll write 4 squared here because each time we're gonna multiply times 4 again so the next is gonna be 4 to the third because we're really each iteration each side turns into 4 sighs that's where that came from 4 squared we lose we're factoring out the square root of 3 we're factoring out the 4 we're factoring out the s squared and all we're left is 1 over 3 to the third power squared so times 1 over 3 to the third power squared and we're just gonna keep going like that forever keep going like that forever so on each step we are incrementing we're multiplying by 4 and we're also multiplying or I guess we say the power of this 4 is incrementing so as close to goes from there's really for the zeroth power here we have a 1 here you can kind of imagine implicitly the 4 to the first power for a square that'll go 4 to the third and then we have this power is also incrementing 3 to the first 3 second 3 to the third but we see that this power is always 1 more than that and it'll be much easier to calculate this infinite which is going to turn into an infinite geometric series if those are actually the same power so what I want to do is I want to increase the power 4 and all of those but I can't just really nearly multiply everything by 4 if I'm gonna multiply everything by 4 I also need to divide everything by 4 so what I'm gonna do in this step right over here I'm gonna multiply and the everything by four so if we divide by four I can do that on the outside so let me multiply it 1/4 times this right over here and so I'm dividing by four out here then I'm going to multiply this by four it so I'm not going to be changing the value of the actual thing so this is going to be this is going to be 4 plus 3 times 4 3 times 4 plus 3 times 4 squared 4 to the third and so what was cool about this is now that the power of 4 and the power of this 3 down here are going to be the same power but it still seems a little weird because we're taking this 1 over 3 square and then we're scaring it 1 over 3 to the 3rd and then we're squaring it and here we just have to realize so this is always going to be squared and this is the thing that's incrementing but in general in general if I have 1 over 3 to the N and I'm squaring it this is equal to 1 over 3 to the 2n power which is so I'm just multiplying it by 2 all right if I'm raising something to exponent then raising that to an exponent that's just multiplying it times the or raising it to the N times 2 exponent and this is the exact same thing as 1 over 3 squared raised to the nth power so we can actually switch these two exponents in a very legitimate way and then let me rewrite everything because I don't want to do too much on this one step right over here so this part right over here gives us square root of 3 s squared over 16 and then that's going to be times I'll open and close the parentheses so then we have 4 plus then in blue I'll write 3 times 4 to the first power and then I can write this I can rewrite this as x 1/3 we could view this as 1/3 squared or we could view this we could views this 1/3 to the 1 over 3 to the 1st power squared or we could view this is 1 over 3 squared to the first power and I'm gonna write it that way so times 1/9 to the first power and then plus 3 times 4 squared and then this we can write as times 1/9 to the 2nd power and then and this one we can write plus three times four to the third times times and this is we could write this as one over twenty-seven to the second power but we could also write this based on what we saw over here we could write this as one over three squared to the third power based on this right over here let me make this clear 1 over 3 to the third the second power this is the same thing as 1 over 3 squared to the third power that's what we showed right over here so this is the quick equivalent to 1/9 one ninth to the third power and now we start to see the pattern it's starting to clean up it's starting to clean up a little bit and let me just do one more step and then we'll finish this in the next video so this is equal to square root of 3 s squared over 16 times 4 plus 4 plus 3 times this is 4/9 plus the next term is 3 times 4/9 squared 4 9 squared and then we have plus plus 3 times 4 over 9 to the third power and then we're just going to keep going on and on and on and on taking 3 times 4/9 to the successively larger and larger powers so this is what we have to find the sum of to find our area and we're going to do that in the next video we're going to use some of the tools we've used to find the sums of infinite geometric series but we're kind of going to redo it in this video just so that you don't have to remember that that formula or that proof