A ball just fits inside a closed cylindrical tin, touching the curved wall and both flat ends. Find the ratio of the surface area of the ball to the curved surface area of the tin.

The tin has the ball's radius $r$ and height $2r$. Ball = 4π$r^2$; curved surface of the tin = 2π$r\times 2r=4$π$r^2$. Ratio = 1 : 1 — they are exactly equal. (2 : 3 compares the ball with the tin's TOTAL surface 6π$r^2$, and is also the ratio of their volumes; 3 : 2 reverses that; 1 : 2 uses 2π$r^2$, the curved surface of a hemisphere, for the whole ball.)