The diagonal section of a regular quadrangular prism has an area Q. Find the area of the lateral surface of the prism.

Since the prism is correct, there is a square at its base, and the side faces are equal rectangles. Let AB = BC = СD = AD = X cm, AA1 = BB1 = CC1 = DD1 = Y cm.
From the right-angled triangle ABC, according to the Pythagorean theorem, AC^2 = X^2 + X^2 = 2 * X^2.
AC = X * √2 cm.
Then Ssec = AA1 * AC = Y * X * √2 = Q cm2.
X * Y = Q / √2 is the side face area.
Then S side = 4 * Q / √2 = 2 * √2 * Q cm2.
Answer: The lateral surface area is 2 * √2 * Q cm2.