Show that apothem of the equilateral triangle, of which one side length is a, will be sqrt3/6a?

Show that apothem of the equilateral triangle, of which one side length is a, will be sqrt3/6a.

1 Answer
Dec 11, 2015

Delta OAB is isosceles.

Explanation:

Delta ABC is equilateral. OH is the apothem. H in AB.

In Delta OHA, hat O = (120º)/2 = 60º, hat H = 90º

|OH| = x ; |OA| = r, |AH| = a/2

tan 60º = frac{|AH|}{|OH|} = (sqrt 3 / 2) / (1 / 2) = (a/2) /x

sqrt 3 = a / (2x)

x = a / (2 sqrt 3)