Area of equilateral triangle

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.