Slicing a Sphere along Two Parallel Planes

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In their book Calculus, One and Several Variables [1], Salas and Hille mention "an interesting property of the sphere" and propose the following exercise:
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Contributed by: Erik Mahieu (November 2015)
Open content licensed under CC BY-NC-SA
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Reference
[1] S. L. Salas and E. Hille, Chapter 10.10, problem 25, Calculus: One and Several Variables, 6th ed., New York: John Wiley and Sons, 1990.
Permanent Citation
"Slicing a Sphere along Two Parallel Planes"
http://demonstrations.wolfram.com/SlicingASphereAlongTwoParallelPlanes/
Wolfram Demonstrations Project
Published: November 24 2015