The sphere is “inscribed” in the cylinder. It’s north pole just touches the top of the cylinder; the south pole just touches the bottom. And the cylinder and sphere just barely make contact all along the equator. If the cylinder were the least bit shorter or skinnier, the sphere would not fit inside.
If r = the radius of the sphere and the radius of the cylinder, then 2r = the height of the cylinder, Vc = volume of the cylinder =(¼r2)(height) = 2¼r3, and Vs = volume of the sphere = (4/3)¼r3. Therefore, Vc/Vs = 3/2, the ratio Archimedes was so proud of. The can will hold 50% more soda pop than the ball.
It is interesting to note that the ratio of the cylinder’s area to the sphere’s area is the same as the ratio of their volumes: Ac = area of top + area of bottom + area of side = ¼r2 + ¼r2 + (2¼r)(2r) = 6¼r2 and As = 4¼r2, so Ac/As = 3/2. It takes 50% more paint to decorate the can than to decorate the ball.
The area of the side of the cylinder (ignoring the top and bottom) is 4¼r2, the same as the area of the sphere. The same amount of gift paper is required to wrap the tube and the ball.
Last Updated February 7, 2022