What’s the density of the black hole in galaxy M87?

On page 20 of the March 2004 issue, you say the 3-billion-solar-mass black hole in the center of the galaxy M87 has an average density about that of air. I would love to understand how this makes any sense.

M87

This zoom into M87, a giant elliptical galaxy with a 4-billion-solar-mass black hole, shows the black hole-powered jet (blue).
NASA / The Hubble Heritage Team

The diameter of a black hole scales directly with its mass. That is, if you double the mass, you double the hole’s diameter. But if you double the diameter of something, you increase its volume by 2^3,or 8 times. So with twice the mass but eight times the volume, you’ve reduced the average density of your black hole by dumping stuff into it. The situation is complicated by the weird distortion of space-time at a black hole, but we can gloss over that for simplicity’s sake.

A 1-solar-mass hole has a diameter (twice the “Schwarz-schild radius”) of 6 kilometers (4 miles). So a billion-solar-mass hole has a diameter of 6 billion kilometers — and an average density only a billionth of a billionth that of the 1-Sun hole.

— Alan M. MacRobert