Why is the apothem of an equilateral triangle one third its height?

Answer 1

The apothem of a regular polygon is the distance from the centre to one of its sides.

Let the equilateral triangle (regular polygon of three sides) be ABC, centred at O. Drop a perpendicular from O to D on side AB.

Consider triangle OAB, ∠OAB and ∠OBA are each 30°.

Since sin(30°)=1/2, it is clear that OD = OA/2 = OC/2

Therefore
Apothem OD=(OD+OA)/3=CD/3=height/3.