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Video transcript

let's say have an equilateral triangle where the length of each side is 14 so this is an equilateral triangle all of the sides have length 14 and inside that I have another equilateral triangle right over here where the length of each of the sides is 4 and what I'm curious about is the area of the region let me color this in a different color is the area of the region that I'm shading in right here so it's the area inside the larger equilateral triangle but outside of the smaller equilateral triangle so let's think about how we would do this I encourage you to pause this and try this on your own well the shaded area shaded area is going to be equal to the large equilateral triangles area so large equilateral triangle area minus the area of the small equilateral triangle minus the small equilateral triangle area so we just have to figure out what the area of each of these equilateral triangles are and so to do it we remember that the area of a triangle is equal to area of a triangle is equal to 1/2 base times height but how do we figure out the height of an equilateral triangle so for example if I have an equilateral triangle like this let me draw it big so I can dissect it a little bit so that would equal valve trying an equilateral triangle like this the length of each of the sides are s and I always have to reprove it for myself just because I always forget the formula we remember that the angles are 60 degrees 60 degrees and 60 degrees they're all equal and what I like to do to find out the area of this in order to figure out the height is I drop an altitude so I drop an altitude just like here and it would split the side into I know it doesn't look like it perfectly because I didn't draw it to scale but it would split it in two it would form these right angles and what's neat about this is I've now split my my equilateral triangle into two 30 30 60 90 triangles 30 60 90 trying and that's useful because I know the ratio of the sides of a 30-60-90 triangle if this is s and I've just split this into this orange section this orange section right over here is going to be s over 2 this is also going to be s over 2 right over here they obviously add up to s and then we know from 30-60-90 triangles that the side opposite the 60-degree side is square root of 3 times the shortest side so this altitude right over here this altitude is going to be square root of 3 s over 2 and now we can figure out a generalized formula for the area of an equilateral triangle it's going to be equal to 1/2 times the base well the base is going to be s so the base is s and the height is square root of 3 s over 2 square root of 3 square root of 3 s over 2 and so this will simplify to let's see we have in the numerator we have a square root of 3 s squared over 4 over 4 and now we can apply this to figure out the areas of each of these triangles so this is going to be equal to the area of the larger triangle is going to be square root of 3 over 4 times 14 squared and the area of the smaller triangle is going to be square root of 3 over 4 times 4 squared and let's see we could factor out a square root of 3 over 4 so this is going to be equal to square root of 3 over 4 times 14 squared 14 squared minus 4 squared minus 4 squared which of course we know this to be is 16 but now let's Jack Chua we evaluate this to actually get a number here and I could try to simplify it by hand but instead let me actually just get my get my actually let's just let's just simplify it by hand first so in case you haven't memorized your 14 Tiny's tables we could just work that out 14 times 14 4 times 14 4 times 4 is 16 and then carry the 1 4 times 4 is 1 4 plus 1 so it's 56 then you put a 0 right there so because we're multiplying by 10 now 10 times 14 is 140 and so this is 196 so this is equal to this is equal to square root of 3 over 4 times 196 196 minus 16 minus 16 which is equal to 180 so this right over here is equal to 180 and 180 is divisible by 4 so this is going to be equal to the square root of 3 times let's see 180 divided by 4 is going to be 45 so it's going to be 45 square roots of 3 did I do that right 180 divided by 4 times 45 is 160 plus 20 is exactly 180 so it's 45 square roots of 3 and if I wanted to get an approximate answer as a decimal so let me get my calculator out so let's go 45 times the square root of 3 would get us 277 if I wanted to round to the nearest hundredth say that seventy seven point nine four so this is approximately equal to 77 point nine four square units the area of the shaded region