If you allow the use of area elements, yes, there's a proof. Circumscribe a cylinder about the sphere of unit radius tangent at the equator. Project the sphere outward to the cylinder from the axis, i.e. the lines of projection are parallel to the equatorial plane. An area element at latitude λ is located distance cosλ from the axis, so it gets expanded by projection by factor 1/cosλ. But the normal to the element on the sphere is inclined to the line of projection by angle λ, and so it gets foreshortened by factor cosλ. So projection preserves area, independent of latitude. -- Gene
________________________________ From: Dan Asimov <dasimov@earthlink.net> To: math-fun <math-fun@mailman.xmission.com> Sent: Subject: Re: [math-fun] Gary Antonick is edging away from the following bonus puzzle
Gene, is there a rigorous proof of that using just Eucidean solid geometry, without calculus?
--Dan
----- That z is uniform in [-1,+1] is a consequence of the solid geometry theorem that the area of a sphere between two parallel planes depends only on the separation between the planes, and is independent of what part of the sphere lies between the planes.