How do you find the distance from the center of an equilateral triangle, with the side length of L, to a point?

Answer 1

The centre of an equilateral triangle is the centroid of the triangle.
That centroid is 2/3 of the length of the median away from a vertex.
So we need the length of a median.

Drawing one median in an equilateral triangle results in right angled triangle with angles 30-60-90, and you should know that the sides of that triangle are in the ratio of 1:√3:2
So if the hypotenuse is L, then
median/L = √3/1
median = √3 L

so the distance from the ''centre" to the vertex is
(2/3)(√3 L) = 2L√3/3