Technically, the radius of a black hole is uniquely determined by its mass, so one can construct a unit system where mass and radius use the same unit of measure...

I want to see that: "How many solar masses is it from Paris to London again?" (115... it's actually a kind of convenient unit, quite close to half an English league and just a bit larger than a Roman league.)

Or: "Man! I stepped on the scales today and I've put on another ten million Planck lengths!"

Randall has us covered, as always.

http://xkcd.com/687/

and

"You can even plug it into Wolfram|Alpha, and it’ll tell you that 20 MPG is about 0.1 square millimeters (roughly the area of two pixels on a computer screen)."

https://what-if.xkcd.com/11/

Wording nit: entirely determined, not uniquely determined. The former says that mass => size. The latter says that a black hole's mass-size relationship is necessarily unique. Which I don't believe you intended.

That unit system might allow comparing black holes to other black holes. But our sun does not have the same density as a black hole, or? The nature article [1] does not compare it to the size of our sun, but to its mass.

[1] http://www.nature.com/nature/journal/v518/n7540/full/nature1...

Actually, you can use this unit system (called Geometrized units [1]) for anything, but it is especially convenient for describing black holes and general relativity in general. As a rule of thumb, in the geometrized units, 1 solar mass = ~1.5 km = ~5 microseconds. They have the same units so you can use them interchangeably (Our sun weights 5 microseconds).

I'm just being pedantic here obviously..

[1] http://en.wikipedia.org/wiki/Geometrized_unit_system

Actually, black holes density decreases linearly as their mass increases. So, there could easily be a black hole with a density of the sun with supper massive black holes having a density near that of water.

(Where density = mass / volume inside event horizon). Because Gravity ~= Mass/distance ^ 2 but volume is 4/3 pi * r ^ 3