Archimedes' Hat-Box Theorem


Enclose a sphere in a cylinder and cut out a spherical segment by slicing twice perpendicularly to the cylinder's axis. Then the lateral surface area of the spherical segment S_1 is equal to the lateral surface area cut out of the cylinder S_2 by the same slicing planes, i.e.,


where R is the radius of the cylinder (and tangent sphere) and h is the height of the cylindrical (and spherical) segment.

See also

Archimedes' Problem, Cylinder, Sphere, Spherical Segment

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Cundy, H. and Rollett, A. "Sphere and Cylinder--Archimedes' Theorem." §4.3.4 in Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., pp. 172-173, 1989.

Referenced on Wolfram|Alpha

Archimedes' Hat-Box Theorem

Cite this as:

Weisstein, Eric W. "Archimedes' Hat-Box Theorem." From MathWorld--A Wolfram Web Resource.

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