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Area of equilateral triangle (advanced)

Sal finds a shaded area defined by two equilateral triangles whose side lengths are given. Created by Sal Khan.

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Video transcript

Let's say I 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. Now 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. And I encourage you to pause this and try this on your own. Well the shaded area is going to be equal to the large equilateral triangle's area minus the area of the small equilateral triangle. 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 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 little bit-- so I have an equilateral triangle like this. The length of each of the sides are s. And I always have to re-prove 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 in two. 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 equilateral triangle into two 30-60-90 triangles. 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 in two, this orange section right over here is going to be s/2. This is also going to be s/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, is going to be square root of 3s/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 3s over 2. And so this will simplify to, let's see we have in the numerator we have the square root of 3s squared over four. 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 3/4 times 14 squared. And the area of the smaller triangle is going to be square root of 3/4 times 4 squared. And let's see, we could factor out a square root of 3/4. So this is going to be equal to square root of 3/4 times 14 squared minus 4 squared. Which of course we know is to be, is 16. But now let's actually 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-- Actually, let's just simplify it by hand first. So in case you haven't memorized your 14 times tables, we could just work that out. 14 times 14, 4 times 14. 4 times 4 is 16. And then carry the one. 4 times 4 is 4 plus 1, so it's 56. You put a 0 right there, because we're multiplying by 10 now. 10 times 14 is 140. And so this is 196. So this is equal to square root of 3/4 times 196 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 to 77. If I want to round to the nearest hundredth, say, that's 77.94. So this is approximately equal to 77.94 square units. The area of the shaded region.