How black hole size scales with mass
Why the event horizon grows in a straight line with mass, from stellar black holes to Sgr A* and the giant M87*, using r = 2GM/c^2.
A straight line, not a curve
The Schwarzschild radius r = 2GM/c^2 is linear in mass, which sets black holes apart from ordinary bodies. A star or planet held up by pressure shrinks only slowly as you add mass, because gravity and internal support fight each other. A black hole has no such support, so its horizon grows in exact proportion to its mass. Double the mass and the radius doubles, ten times the mass gives ten times the radius, with no diminishing return.
From stellar to supermassive
Plugging numbers in shows the enormous range. One solar mass yields a horizon of about 2953.99 metres, roughly the width of a small town. Sagittarius A*, the four point three million solar mass black hole at the centre of the Milky Way, comes out to about 1.27022e10 metres, wider than the orbit of Mercury. M87*, the six point five billion solar mass giant famously imaged in 2019, reaches about 1.92010e13 metres, larger than the entire Solar System. All three follow the same simple straight line.
Why the density falls as they grow
Because radius scales with mass but volume scales with radius cubed, the average density inside the horizon falls sharply as a black hole grows. Mass rises like r while the volume it occupies rises like r cubed, so density drops like one over mass squared. A stellar black hole is denser than an atomic nucleus, yet a supermassive one can have a mean density below that of water. Size, not crushing density, is what makes the largest black holes so vast.
What the horizon is and is not
The Schwarzschild radius marks the event horizon, the one way boundary from which no signal can return, not a solid surface. Crossing it requires no special local sensation for an infalling observer, yet from outside nothing that passes it is ever seen again. The formula assumes a perfectly spherical, non-rotating, uncharged mass, so real astrophysical black holes, which spin, have a slightly smaller horizon described by the Kerr solution. Even so, r = 2GM/c^2 gives an excellent first estimate of any black hole's scale.