*Problem #35*

A *Reuleaux triangle* is constructed by taking an equilateral
triangle *ABC* and drawing the three circular arcs: *BC* with
center *A*, *AC* with center *B*, and *AB* with
center *C*, as shown below. The Reuleaux triangle is an example of a
"curve of constant width".
This month's problem is to find the volume and the surface area of the
solid obtained by rotating the Reuleaux triangle shown above around a
vertical axis passing through vertex *A*. Express your answer in
terms of *r*, the length of a side of the original equilateral
triangle.