As an engineer by schooling, my first thought was this question has been around since the first underground tank was installed and there is a straightforward formula written down somewhere.
I googled 'volume cylindrical segment' and came up with what I posted earlier. I made no claim that this was my original thinking and derivation. I have no desire to debate the higher math involved. There was an excellent little graph included with the discussion that I have posted here.
If you highlight the graph the numbers show up a lot better. Sorry.
It's nice to be important,
but it's more important to be nice.