Equable Spherical Segment

EQUABLE SPHERICAL SEGMENT

Balmoral Software

Solutions: 0

Let R be the radius of a sphere and h the height of a spherical segment of one base, where 0 < h < 2R. The volume of the spherical segment is V, where

V/π = h²(R - h/3)

The lateral surface area of the spherical segment is S_l, where

S_l/π = 2Rh

The surface area of the base is S_b, where

S_b/π = r²

and r is the radius of the base. The above three quantities are integers by convention. The base radius is related to the sphere radius and segment height by

r² = R² - |R - h|² = 2Rh - h² = S_b/π,

S_l/π - S_b/π = h²

is an integer and therefore

is rational. Then

S_l/π - 2h²/3 and 2(R - h/3)

are both rational, so

is also rational, and it follows that h is an integer.

The equability requirement is

V/π = S_l/π + S_b/π
h²(R - h/3) = 2Rh + 2Rh - h²

[1]

We have

4 < h < 6,

so h = 5. It follows from [1] that R = 10/3. But then S_b/π = 25/3 is not an integer, so there are no equable spherical segments with integer volume and surface area sections.

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