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## Geometry (all content)

### Course: Geometry (all content) > Unit 7

Lesson 9: Advanced area with triangles# Area of equilateral triangle

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

## Want to join the conversation?

- Couldn't we use the trig functions to find this too? We have all the angles, so we could have found the height using one of the sides of the Triangle, then plugged that into the area formula.(31 votes)
- what if the equalateral triangles has unequal sides?(0 votes)
- Well, then, it won't be equilateral! An equilateral triangle is, by definition, a triangle with all equal sides. It would be a scalene or isosceles instead.(11 votes)

- How do you figure out the height of an equilateral triangle?(3 votes)
- First draw the height - due to the symmetry of equilateral triangles, it will start at the midpoint of whichever side you choose to start from, and end at the opposite vertex (point/corner). In effect you will split the equilateral triangle into two congruent right triangles.

Now consider one of these two right triangles by itself. If the equilateral triangle has sides of length x, then the hypotenuse of our right triangle will also be x. We also know that the side opposite the 30 degree angle has length x/2, since we split that side of the triangle in half to construct this right triangle. Since you know two of the sides of a right triangle, you can use the Pythagorean theorem to find the length of the 3rd.

(x/2)^2 + m^2 = x^2

x^2/4 + m^2 = x^2

m^2 = (3*x^2)/4

m = (x*sqrt(3))/2

Where m is the height of the right triangle, which is equal to the height of the equilateral triangle. Incidentally, this derivation also proves the shortcut for the ratio of the sides in a 30-60-90 triangle, since the effect of cutting an equilateral triangle in half is to create 2 30-60-90 triangles.(5 votes)

- Isn't it easier to just multiply the 1/2 with the base and height rather than to just go through all of those square roots because he lost me there(1 vote)
- We don't know what the height is without working it out. Pythagoras' theorem says that in a right triangle, the area of a square drawn on the longest side (the hypotenuse) is equal in area to the areas of the two squares drawn on the other two sides (the legs), added together. The square on the side with length s/2 is 1/4 of the area of the square on the side with length s, so a square on the height must be 3/4 of it. This tells us that the height is s times the square root of 3/4, which is equal to s times a half of the square root of three.(6 votes)

- what does he mean by 30,60,90 video(3 votes)
- 2:03Why is the square root of 3 used? I Don't see how it came up. I don't understand that bit. Help me please.(1 vote)
- The √3 comes from a special property that applies only to 30°-60°-90° right triangles. For just that type of triangle, if the hypotenuse is h, then the shortest side (the side opposite the 30°) will have a length of ½ h. The second longest leg, the one opposite the 60° will have a length of ½ h√3 .

Another way of saying the same thing is that the ratio of the sides of a 30°-60°-90° are 1L : L√3 : 2L where L is the length of the shortest side, the side opposite 30°. L√3 is the length of the second shortest side and 2L is the length of the hypotenuse.

If you cannot remember this, don't worry, you can always do solve it with the usual sin, cos, tan functions.(3 votes)

- One of the sides of an equilateral triangle is 22cm and the height is 16 cm what is the area(1 vote)
- This is not a possible situation. It follows from the 30-60-90 right triangle side length ratio that the ratio of the height to the side length of an equilateral triangle is sqrt(3) / 2, or approximately 0.866. The ratio 16/22 is approximately only 0.727.

Have a blessed, wonderful day!(3 votes)

- Can you use the herons formula for equilateral triangles?(1 vote)
- Yes, but I would suggest using the formula Sal gave:

3s² / 4

where 's' is any one side of an equilateral triangle.

The reason why is because it saves a lot of time and effort. In fact, I have never heard of Heron's formula until now, but the formula I provided above is one of the easiest formulas to memorize AND use in my opinion. However, you can always use the Heron's formula if that is easier for you.(3 votes)

- Isn't the area of a triangle formula just "base x height over two"? If so, why do we need a specific formula for equilateral triangles?(2 votes)
- It's useful if we don't know the altitude length of the equilateral triangle and we only have its side length. From the perspective of math competitions, you can calculate equilateral triangle area with this video's formula many times faster than the standard bh/2.(2 votes)

- I really dont get that, is 30 a random number:((1 vote)
- The 30 degree angle is found from taking the 60 degree triangle and slicing it directly in half.(2 votes)

## 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.