Solution

Usually, you either get it straight away or you don’t get it at all.

The area of the base (B) is easy (remember to allow for the thickness of the glass). Tip a random amount of champagne out leaving enough so that, when you turn the bottle upside down, the liquid level is above the tapered neck and well into the main cylinder. With the bottle right way up, measure the height of the liquid above the base (h₁). When you upend it the air occupies a different space but it cannot be any more or less than the original volume. Measure the height below the base (h₂). The total volume must be the volume of champagne plus the volume of air so the total volume is B*(h₁+h₂).