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V/π = hThe lateral surface area of the spherical segment is S^{2}(R - h/3)

SThe surface area of the base is S_{l}/π = 2Rh

Sand 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_{b}/π = r^{2}

rso^{2}= R^{2}- |R - h|^{2},

It follows that h

is rational. Then

is also rational. The fraction on the right is a quotient of two rational numbers, so 2h is rational and it follows that h is an integer.

The equability requirement is

Since h < 2R, we have

[1]

so h = 5. It follows from [1] that R = 10/3. But then S4 < h < 6,

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