If you want to put a 4x8 piece of plywood through a 3 foot square opening in your ceiling by turning it diagonally is the opening big enough? use a 45-45-90 since its a square

Yes, the 3 foot 3 foot square opening in your ceiling is big enough to insert a  4x8 piece of plywood.

In  a 3x3 foot square, we can form two right triangles with same diagonal side or hypotenuse. .

Using the Pythagorean theorem/formula, we can find the length of the hypotenuse or the longest side of the triangle ( c) given the two sides a and b

Substitute values a=3 and b=3 to formula

a^2^+b^2^=C^2^=

3^2^+3^2^=C^2^=

9+9= C^2^

C= square root of 18

C=4.24 

A 4x8 piece of plywood can be inserted in a 4.24 opening diagonally.