Improving on Archimedes: New Proofs of the Volume and Area of Sphere and Cylinder

Archimedes published On the Sphere and Cylinder around 225 B.C. It derived formulæ for the area and volume of the sphere and cylinder, and proved that the area of a sphere is equal to that of the circumscribed cylinder (minus the end caps), which is the basis of the Lambert cylindrical equal-area map projection, much beloved by those rendering models of celestial bodies from spacecraft imagery. Is it possible, more than twenty centuries later, to find novel derivations, and what new insights do they provide?