The volume of a regular tetrahedron is very easy to calculate (cube half the length of a side) but once it is inflated and pressurised (even at the low pressures found inside a hot air balloon) the volume calculations become more complex.
Raven industries were commisioned in 1978 to determine the volume of an inflated tetroon and published this reasearch paper as it findings. They used experimental methods to determine approximate volumes of 3 different sized tetrahedrons at a variety of superpressures. In summary they found that the volume of a pressurised tetroon is between 3 and 6 percent greater than than of a regular tetrahedral pyramid of the same side length.
More modern papers can be found on the web by searching on "Spherical Tetrahedron Volume"
Download: Raven Tetroon Volume Report