Math riddle

A buddy and I were pondering this "riddle" today. I dreamed it up, but
our combined expertise was not enough to crack it, though we both had a
good deal of college-level math. I have a gut feeling that there is a
solution that goes no farther than Algebra & Trigonometery - not
requiring calculus - but I may be wrong.

Suppose you are walking on a perfecly flat plane.
You drive a stake into the ground and attach a 50 mile rope to it.
You walk 50 miles, until the rope is taut.
Then you walk a circle around the stake, keeping the rope taut.
The distance you will walk - discounting the imprecision of a human
walking, and the sag introduced by the rope's weight - will be 100 * pi
miles.

Ok? Now part 2:

Suppose you are walking on the surface of a sphere (as we more-or-less
are).
You drive a stake into the ground and attach a 50 mile rope to it.
You walk 50 miles until the rope is taut.
Then you walk a circle around the stake, keeping the rope taut.
The distance you walk will be less than 100*pi miles, because of the
curvature of the sphere's surface. The 50 mile radius is not a
straight line, it's curved, and thus you are actually less than 50
linear miles from the stake.

NOW, HERE'S THE QUESTION:

Assuming perfect measurements, and using only your radius , your
measured circumference and the constant pi, can you compute the size
of the sphere you are walking on?

If anyone can think of a solution, I'd love to hear it.

.

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