There are two things that you need to know, in order to deduce the formula. 1. Equilateral triangles have equal sides and angles of 60 degrees. And, 2. you should know the side ratios of 30-60-90 triangles. Lets look at the latter first
The shortest side of a 30-60-90 triangle is equal to s. The side perpendicular to the shortest side is equal to Sqrt(3)*s. The longest side (hypotenuse) of a 30-60-90 triangle is equal to 2s.
You can convince yourself of the validity of these statements using Pythagoras' Theorem.
Now, The area of any triangle is equal to half the product of the length of its base and height. Its base is easy, its the length of any of its sides. Lets call the length of the side of a equilateral triangle K. Now all that is left is to determine the height. If we draw an altitude from the highest vertex to the base we see that we have a 30-60-90. The height of it is the side perpendicular to the base. the length of the leg that is perpendicular to the height is equal to K/2, so according to our ratio the height must be equal to Sqrt(3)* (K/2).
So the area of a equilateral triangle is equal to (1/2)bh = (1/2)(K)[Sqrt(3)*(K/2)] = (1/4)[K^2 * Sqrt(3)], where K is the length of a side, * means multiplication, and ^ means exponentiation.
Hope this helped. Its always better to know how to derive a formula than to just memorize it. Understanding its derivation means you really understand where it comes from and how to use it quickly.
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