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Equal Cores and Shells in Circles and Spheres

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The area of the red circular ring between the radii and is given by . This is equal to the area of the blue disk of radius given by if the three radii satisfy . Visually, the equality of the areas of the shell and disk is often not very obvious, which might loosely be classed as an optical illusion.

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The 3D analog compares the volume of a red spherical shell and a blue central sphere. Recall that the volume of a sphere of radius is given by . The relation between radii is now given by . The spheres are shown in transparent hemispherical cross section.

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Contributed by: S. M. Blinder (July 2013)
Open content licensed under CC BY-NC-SA

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Integer triplets , , such that such as , , , and so on are the well-known Pythagorean triplets. By the Fermat–Wiles theorem (formally known as Fermat's last theorem), there are no analogous integer triplets for cubes.

Snapshot 1: equality of the areas of the shell and disk is not visually obvious

Snapshot 2: these radii belong to the simplest Pythagorean triplet , scaled by factor of 4

Snapshot 3: when , the circle is bisected azimuthally

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