Let D be the large diagonal, d the smaller diagonal, X the side of the rhombus.
The height of the rhombus is:
H = S / X = D * d / 2 = a * a√3 / (2 * X) = a ^ 2√3 / (2 * X).
S is the area of the rhombus.
We find the side of the rhombus X by the Pythagorean theorem, since the angle between the diagonals is 90 degrees, and the rhombus consists of 4 triangles with legs equal to a / 2 and a√3 / 2.
X ^ 2 = (a / 2) ^ 2 + (a√3 / 2) ^ 2 = a ^ 2/4 + 3 * a ^ 2/4 = a ^ 2.
Therefore, X = a.
H = a ^ 2√3 / (2 * X) = a ^ 2√3 / (2 * a) = √3 * a / 2.
Answer: the height of the rhombus is √3 * a / 2.
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