How do you get 4"?
If you say the join should be 2", then the circles going away from it will both move further out on the board, so you only need to make sure the outer circle doesn’t go outside the board at the midpoint between joints. The radius is perpendicular to the board at that point, so you get a right triangle between the inside of the joint, the midpoint between joints and the center, with a 22.5 angle at the center of the circle.
If the join is 2" to the outer circle and the radius of that circle is 45" the hypotenuse for that triangle is 43" and the distance between center and midpoint of the board is
43*cos 22.5 = 39.73
Since the outer circle needs to be 45" the board has to be 45-39.73 = 5.27" wide.