This question is related to the Extending the length of a curcumference by 1 meter problem.
But instead of making a larger circle as in the example above, the loose string is hung from a nail on a wall - like a circular picture frame.
Consider a taut string around the circumference of a cylindrical picture frame of radius R. Then add 1m to the string and put the string around the cylinder and then "hang" the cylinder like a picture frame from a nail on a wall. The string will form an apex at the nail, with the two tangents from the cylinder which then go around the major arc of the circle.
Question: what is the height of the apex from the top of the circle?