The volume of a standard ASME head with dish radius equal to 100% of vessel diameter and knuckle radius equal to 6% of vessel diameter is V = 0.0809 * D^3.  

For a 2:1 elliptical head, the volume is V = Pi / 24 * D^3.  

For a hemispherical head, the volume is V = Pi / 12 * D^3.  

In all of these formulas, use consistent units.  In other words, if diameter is in inches, then the volume is in inches cubed.

An ellipsoidal or 2:1 ellipsoidal head follows a true ellipse.  A 10 % torispherical head profile is almost a true ellipse see  http://www.gowelding.btinternet.co.uk/ped/ped3b.htm
 For a diagram of these two head profiles.

This page shows how the volume for a true ellipse is derived http://www.gowelding.btinternet.co.uk/calcs/indexc.htm

The address of the this sites main entrance is www.gowelding.com