If you know about circles, I would be very surprised if you did not know that the area of a square of side length s is and its 'circumference' (more commonly called 'perimeter' for everything except circles) is P= 4s. You also should know (using the Pythagorean theorem), that the length of a diagonal is and so what you are calling the 'radius' is half the diagonal, . From that, . Replacing s by that in the previous formulas, the area is and the perimeter is .
Notice that the constants multiplying in the area in the perimeter are not the same. There is no single number like for squares.