Question

How do I express the area and perimeter of an equilateral triangle as a function of the triangles side x?

Answer 1

For side x of an equilateral triangle, A = x^2 sqrt(3) / 4

Split the triangle into two congruent triangles with an altitude from one vertex, which will be the height of the equilateral triangle.

This will create two 30-60-90 triangles with base x/2, hypotenuse x, and altitude x/2 * sqrt(3)
from the properties of a 30-60-90 triangle (sides in ratio of 1-2-sqrt(3))

Now for the original triangle, we have base x and height x/2 * sqrt(3)

A = 1/2 (b) (h) = 1/2 (x) (x/2 * sqrt(3)) = x^2 sqrt(3) / 4