The diagonal of a regular quadrangular prism is d and makes an angle β with the plane

The diagonal of a regular quadrangular prism is d and makes an angle β with the plane, find the area of the diagonal section.

In a regular prism, the diagonals are equal and at the point of their intersection they are divided in half, then the AOC triangle is isosceles, and then the angle OAC = OCA = β0.

Then the angle AOC = (180 – 2 * β).

The diagonal section of the prism is the rectangle АА1С1С, then Ssection = (А1С ^ 2 * SinAOC) / 2 = (d2 / 2) * Sin (180 – β) = (d ^ 2/2) * Sin (180 – 2 * β) = (d ^ 2/2) * Sin (2 * β) cm2.

Answer: The area of the diagonal section is (d ^ 2/2) * Sin (2 * β) cm2.