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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.

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.