"The Penguin Book of Curious and Interesting Puzzles", by David Wells...

This is problem 395 from the book:
Consider a vertical girl whose waist is circular, not smooth, and temporarily at rest. Around her waist rotates a hula hoop of twice its diameter. Show that after one revolution of the hoop, the point originally in contact with the girl has travelled a distance equal to the perimeter of a square circumscribing the girl's waist.

And the book's solution:
Since motion is relative, consider the hoop as fixed and the poor girl whirling around. The original point of contact on the girl traverses the diameter of the hoop twice and this is the required distance.

(To understand the solution see the first picture here:
http://www.2send.us/uploads/754a494274.gif )
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Find the error in the answer, and the true distance travelled by the point.

Letting the girl spin comes down to one complete turn of the girl.

At least, that is what I visualisze...

visualize a circle in a circle.
The inner has half the diamater of the outer one.

Now it is unimportant if you rotate the outer one around the inner one,... (Take a coin and make a circle double its size, then rotate the circle by a rolling motion around the coin (you can feel it though the paper)

,... or if you just roll the inner one in the outer one. (roll the coin inside the paper circle)

Clear?