There's a report available on the internet on the acoustic characteristics of St Paul’s Cathedral in London, and it has this little snippet of information: the interior volume of the cathedral is 152,000 m3. Is that a credible number? Let’s use a little bit of solid geometry to do some rough-and-ready cross-comparison.
A quick google at some pictures and measurements tells me that to a very rough approximation, the interior main body of St Paul’s can be approximated by a cuboid, roughly 50m wide, 150m long and 30m high. The famous interior Whispering Gallery is at 30 height and the exterior Stone Gallery around the dome is at 53 metres.
Based on this I don’t think it’s too unreasonable to imagine a simplified shape with the following interior dimensions (if you pushed all the interior stonework to the edges). Width 40m x height 25m x length of 140m, giving a total of 140,000 m3. The dome is around 30m in diameter and together with the cylindrical drum it sits on adds another approximately 30m to the interior vertical height. Working this through gives about another 18,000 m3 for the dome. We’ve reached a total of 158,000 m3 which is enough to convince me that the figure that the acoustic engineers used is probably close enough.
Now for the tennis balls. If you tumble a load of balls into a container, they won’t completely fill the space. If you pack them super-carefully you can bring the proportion of space filled to around 78%, but if you just let them settle for themselves, you can expect around 65% of the space to be filled. A tennis ball of 6.8cm diameter will have a volume of around 165 cm3, but when loosely packed, will occupy a volume of around 250 cm3, roughly a cupful. This means a box with a volume of one cubic metre will hold around 4000 tennis balls (not allowing for the “edge effect” which stops them from being so closely packed around the edges). And that means that the interior of St Paul’s will hold 152,000 times as much for a total of 608 million tennis balls.
But what if, instead of tennis balls, we used pool balls? With a diameter of 5.715 cm, their volume is just about 60% of the volume of a tennis ball. You can see where I’m going with this, can’t you? A metre cubed can accommodate 6700 pool balls, and if we multiply up, we get to 1,018,400 pool balls to fill St Paul’s. And that’s one way to visualise a billion.