TITLE: Tricky 3d geometry problem
QUESTION [15 upvotes]: We have a cube with edge length $L$, now rotate it around its major diagonal (a complete turn, that is to say, the angle is 360 degrees), which object are we gonna get?

Astoundingly the answer is D. And here is a demonstration:

Well now I'm required to calculate the volume of this monster. It's far beyond my capability. I don't know how to analytically describe the curve between the two cones (although I believe it is a hyperbola).  And I'm still not sure why it should be a curve rather than a segment or something.  Could you help me? Thanks in advance.

REPLY [2 votes]: The central portions are hyperboloids of 1 sheet capped by two cones at the ends. ( not catenoids). The hyperboloid of one sheet is a ruled surface formed by rotation of  a skew line about the rotational axis of symmetry.(Examples are cooling towers, hypoid gears etc).
Equations 
Hyperboloid of one single sheet $ \dfrac{(x^2 + y^2)}{a^2} -\dfrac{z^2}{c^2}=1  $ with one negative sign,
Hyperboloid of two separate sheets$ \dfrac{-(x^2 + y^2)}{a^2} +\dfrac{z^2}{c^2}=1 $ with two negative signs.
Volume calculation is done usual way of integration using meridian curve equation given above.