Let ABC be the equilateral triangle with each side equal to a.
Let AD be the altitude from A, meeting BC at D.
Therefore, D is the midpoint of BC.
Let AD be h.
Applying Pythagoras theorem in right-angled ABD, we have:
AB2 = AD2 + BD2
Therefore,
area of triangle ABC = 1/2 × base × height = 1/2 × a × √3/2 a
This completes the proof.