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.


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.


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



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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