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 it 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 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 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 over two and if this length is s over two we can use what we know about 30-60-90 triangles to figure out this side right over here so to figure out what that what the actual altitude is and the reason why I care about the altitude is because the area of a triangle is one half times the base times the height or times the altitude so if this is s over two the shortest side the side opposite opposite the stick opposite the 30-degree angle is s over two then the side opposite the 60-degree angle is going to be square root of three times that so it's going to be square root of three s over two and we know that because the ratio of the sides of a 30-60-90 triangles a 30-60-90 triangle the side opposite 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 is or the hypotenuse is going to be 2 times that so it's 1/2 square root of 3 to 2 so this is a 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 we just need to figure out what what the area of this triangle is using area is equal to 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 figure that out it is square root of 3 times s over 2 and we just multiply that out and we get C in the numerator you get a square root of 3 times 1 times s times s that is 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 its sides were 2 and this would be 2 squared over 4 which is just 1 so we'll just be square root of 3 so we just found a generalizable way to figure out the area of an equilateral triangle