Let ABC, be any equilateral triangle with each sides of lenght 2a. Perpendicular AD is drawn from A to BC.
We know that the altitude in equilateral triangle, bisects the opposite sides.
Therefore, ∴ BD = DC = a
In triangleADB, by Pythagoras theorem
AB² = AD² + BD²
⇒ (2a)² = AD² + a² [Because AB = 2a]
⇒ 4a² = AD² + a²
⇒ AD² = 3a²
⇒ AD = √3a
Hence, the lenght of each altitude is √3a.