Slicing a Sphere along Two Parallel Planes

slice position and size

top: 0.55

bottom: 0.

band surface area: 1.100 π

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

z2

∫

z1

2πf(z)

2

′

f

(z)

+1

dz

, with

f(z)=

1-

2

z

; the integral is

2π(

z

2

-

z

1

)

, which depends only on the difference

(

z

2

-

z

1

)

.