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 
is equal to
the lateral surface area
cut out of the cylinder 
by the same
slicing planes, i.e.,
 |
where 
is the radius
of the cylinder (and tangent sphere)
and 
is the height of the cylindrical (and
spherical) segment.
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Viviani's curve
