A View from the Back of the Envelope top

A ball's volume and area are
1/2 of its box's
On Being Approximate
A sphere's volume and area are easy to remember.

The volume & area of a ball are... 1/2 that of the box it came in.
These approximations are only 5% low.

1/2 the area of a box is 3 sides.
Area is 3d2, or 12r2.
Volume is 1/2 d3, or 4r3.

details.... The volume of a box is Vcube = d3.
The volume of a ball is Vsphere = 4/3 pi r3 = pi/6 d3.
An approximation of 3/6 for pi/6 is 4.5% low.

The area of a box is Acube = 6d2.
The area of a ball is Asphere = 4 pi r2 = pi d2.
An approximation of 3 for pi is 4.5% low.

So unless you care about the 5%, please don't tell children to use 4/3 pi r3...
The amusement of finding pi in an equation no-one remembers is somewhat limited.

A View from the Back of the Envelope
Comments encouraged. - Mitchell N Charity <mcharity@lcs.mit.edu>)