Hello! I will attempt writing this out, but if it ends up being confusing, I can try to create a simple visual (so let me know what you think!).
In theory, if you put a perfect sphere into a perfect cube, the sphere only touches the cube at 6 points (the perfect center of each square side). To find this point of contact, create both diagonal lines from corner to corner for each square side/face. It will create an X. The place where the lines intersect are the point that the sphere is touching the cube!
You can use these points of contact to "connect" the dots and make the circles/ellipses for your sphere.
To find the diameter of the sphere, you would connect the two opposite points of contact where the sphere touches the cube (basically you are cutting it in half by connecting points of contact between sphere and cube). To properly cut this diameter line in half for the radius, you would need to find a diameter line connecting two other points and find where they intersect in the center (basically cutting it in half in the other direction). This will create another X, and where the lines intersect is the center of both cube and sphere!
This sounds like an assignment, so my answer is probably too late to be helpful. However, definitely let me know if it makes any sense or if visuals are better rather than a written answer.
Either way good luck and happy drawing!