Slicing a Sphere along Two Parallel Planes

Initializing live version
Download to Desktop

Requires a Wolfram Notebook System

Interact on desktop, mobile and cloud with the free Wolfram Player or other Wolfram Language products.

In their book Calculus, One and Several Variables [1], Salas and Hille mention "an interesting property of the sphere" and propose the following exercise:


Slice a sphere along two parallel planes that are a fixed distance apart. Show that the surface area of the band so obtained is independent of where the cuts are made.

This Demonstration explores this fact by computing the surface area of the band on a sphere using Mathematica's geometry function Area on a parametrized surface of a sphere.

Moving the sliders up and down shows that the surface area of the band will not change if both sliders are moved simultaneously.

The surface area can also be found by computing the integral , with ; the integral is , which depends only on the difference .


Contributed by: Erik Mahieu (November 2015)
Open content licensed under CC BY-NC-SA




[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.

Feedback (field required)
Email (field required) Name
Occupation Organization
Note: Your message & contact information may be shared with the author of any specific Demonstration for which you give feedback.