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# Area of equilateral triangle

Finding the formula for the area of an equilateral triangle with side s. Created by Sal Khan.

Video transcript

Let's say that this
triangle right over here is equilateral, which
means all of its sides have the same length. And let's say that
that length is s. What I want to do in
this video is come up with a way of
figuring out the area of this equilateral
triangle, as a function of s. And to do that, I'm
just going to split this equilateral in two. I'm just going to drop an
altitude from this top vertex right over here. This is going to be
perpendicular to the base. And it's also going to
bisect this top angle. So this angle is going to
be equal to that angle. And we showed all
of this in the video where we proved
the relationships between the sides of
a 30-60-90 triangle. Well, in a regular equilateral
triangle, all of the angles are 60 degrees. So this one right over here
is going to be 60 degrees, let me do that in
a different color. This one down here is
going to be 60 degrees. This one down here is
going to be 60 degrees. And then this one up
here is 60 degrees, but we just split it in two. So this angle is going
to be 30 degrees. And then this angle is
going to be 30 degrees. And then the other
thing that we know is that this altitude
right over here also will bisect
this side down here. So that this length is
equal to that length. And we showed all of this a
little bit more rigorously on that 30-60-90 triangle video. But what this tells us is well,
if this entire length was s, because all three sides
are going to be s, it's an equilateral
triangle, then each of these, so this
part right over here, is going to be s/2. And if this length
is s/2, we can use what we know about
30-60-90 triangles to figure out this
side right over here. So to figure out what
the actual altitude is. And the reason why I
care about the altitude is because the
area of a triangle is 1/2 times the base times the
height, or times the altitude. So this is s/2,
the shortest side. The side opposite the
30 degree angle is s/2. Then the side opposite
the 60 degree angle is going to be square
root of 3 times that. So it's going to be
square root of 3 s over 2. And we know that
because the ratio of the sides of a 30-60-90
triangle, if the side opposite the 30
degree side is 1, then the side opposite
the 60 degree side is going to be square
root of 3 times that. And the side opposite the 90
degree side, or the hypotenuse, is going to be 2 times that. So it's 1 to square
root of 3 to 2. So this is the shortest
side right over here. That's the side opposite
the 30 degree side. The side opposite
the 60 degree side is going to be square
root of 3 times this. So square root of 3 s over 2. So now we just
need to figure out what the area of this triangle
is, using area of our triangle is equal to 1/2 times the
base, times the height of the triangle. Well, what is the
base of the triangle? Well, the entire base of the
triangle right over here is s. So that is going to be s. And what is the height
of the triangle? Well, we just figured that out. It is the square root
of 3 times s over 2. And we just multiply
that out, and we get, let's see, in the numerator you
get a square root of 3 times 1 times s times s. That is the square root
of 3 times s squared. And the denominator,
we have a 2 times a 2. All of that over 4. So for example, if you have
an equilateral triangle where each of the sides
was 1, then its area would be square
root of 3 over 4. If you had an equilateral
triangle where each of the sides
were 2, then this would be 2 squared over
4, which is just 1. So it would just be
square root of 3. So we just found out
a generalizable way to figure out the area of
an equilateral triangle.