I solved the puzzle in my head before getting to the second paragraph in the article. Here's the reasoning:<p>1) The area of a circle has a fixed ratio to the area of a square inscribed in that circle.<p>2) Therefore the volume of a cylinder has a fixed ratio to the volume of a square box of the same height, which sits inside that cylinder.<p>3) Therefore the biggest cylinder corresponds to the biggest box that can fit inside the sphere.<p>4) That box is obviously a cube, because what else could it be?<p>5) If a cube is inscribed in a unit sphere centered at the origin, the corners have coordinates ±1/√3, ±1/√3, ±1/√3.<p>6) Now you can calculate the volume of the cylinder in your head. Do it!