Black Hole Calculator

Enter a mass in solar masses to explore black hole geometry and thermodynamics — from micro black holes to supermassive quasar engines.

Worked example — Black Hole Calculator

Schwarzschild	29.5 km
Photon sphere	44.3 km
ISCO	88.6 km
Hawking temp.	6.17 nK

These are the values the calculator shows when the page opens. Change any input above to recompute them.

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Nothing that falls in is lost, only scrambled 🖖

In 1974, Hawking showed that quantum effects near the event horizon cause black holes to radiate thermally — as if they were a black body at temperature T_H ∝ 1/M. As the hole radiates, it loses mass and shrinks, eventually evaporating entirely. But here lies the paradox: thermal radiation carries no information about what fell in. Quantum mechanics demands that information is never destroyed — yet Hawking's calculation seems to erase it. This is the black hole information paradox. Decades of debate produced no resolution, until recent work involving 'islands' and the Page curve suggested that information may be encoded in subtle quantum correlations in the Hawking radiation itself — entanglement structure invisible to local observers. The paradox is not fully resolved, but the current consensus is that unitarity survives: information escapes, encoded in a way that requires a complete theory of quantum gravity to decode.

One number sets the whole geometry 🖖

A non-rotating black hole is described entirely by its mass. Once you know the Schwarzschild radius r_s = 2GM/c², every other landmark follows as a fixed multiple: the photon sphere at 1.5 r_s, the innermost stable orbit at 3 r_s. Double the mass and all of them double with it. As a black hole, the Sun would span just ~3 km across; Earth, a mere ~9 mm.

Black holes get hotter as they shrink 🖖

Ordinary objects cool as they lose energy; black holes do the reverse. Because T_H ∝ 1/M, radiating mass away makes a hole hotter, not colder — a negative heat capacity. Evaporation therefore runs away: the smaller the hole shrinks, the faster it radiates, ending in a final burst. It also means a black hole can never settle into stable thermal equilibrium with an unlimited heat bath.

Problem solved in full

The event horizon of a 10-solar-mass black hole 5 steps
Find the event horizon of a 10-solar-mass black hole, then answer something that sounds absurd: how dense does a black hole have to be?
1. Take the escape-velocity result and ask where it equals the speed of light. This is the Newtonian shortcut — a full derivation needs general relativity — and it happens to give exactly the right answer, which is a historical accident worth knowing about: Michell wrote it down in 1783.
  
  $v_{esc} = \frac{2 GM}{R} = c ⟹ R_{s} = \frac{2 GM}{c ^{2}}$
2. Substitute ten solar masses. The calculator above prints this radius, and the striking feature is the proportionality rather than the number: R_s scales linearly with mass.
  
  $R_{s} = \frac{2 ( 6.674 \times 1 0 ^{- 11} ) ( 10 \times 1.989 \times 1 0 ^{30} )}{( 2.998 \times 1 0 ^{8} ) ^{2}} = 2.95 \times 1 0^{4} m = 29.5 km$
3. Two other radii follow as fixed multiples, which is why the tool prints them together: light itself can orbit at 1.5R_s, and no stable circular orbit exists inside 3R_s — that inner edge is what sets the size of an accretion disc.
  
  $r_{ph} = 1.5 R_{s} = 44.3 km, r_{ISCO} = 3 R_{s} = 88.6 km$
4. Now the consequence. Density is mass over volume, and volume goes as radius cubed while radius goes as mass — so density falls as the inverse square of mass. Bigger black holes are less dense.
  
  $ρ = \frac{M}{\frac{4}{3} π R _{s}^{3}} \propto \frac{M}{M ^{3}} = \frac{1}{M ^{2}}$
5. Put numbers on it. A stellar-mass hole is denser than an atomic nucleus. A supermassive one is not.
  
  $M = 10 M_{⊙} : ρ \approx 1.8 \times 1 0^{17} kg/ m^{3} M = 1 0^{8} M_{⊙} : ρ \approx 1.8 \times 1 0^{3} kg/ m^{3}$
Answer

29.5 km, with the photon sphere at 44.3 km and the innermost stable orbit at 88.6 km. But the last line is the one that changes how the object feels: a 10⁸-solar-mass black hole has an average density of about 1800 kg/m³ — a little denser than water, less dense than iron. You could cross the horizon of one without noticing anything locally. “Black hole” names a geometry, not a substance, and the horizon is a place where escape becomes impossible rather than a surface made of anything.

$R_{s} = \frac{2 GM}{c ^{2}} \propto M ⟹ ρ \propto \frac{1}{M ^{2}}$

References (2)

Insight blocks 1 and 3 — the radiation, and the 1/M temperature that makes evaporation run away: S. W. Hawking, "Black hole explosions?" Nature 248, 30–31, 1974.
The curve block 1 names: D. N. Page, "Information in black hole radiation." Physical Review Letters 71(23), 3743–3746, 1993 — where the entropy of the radiation is argued to turn over rather than rise forever.

By the 42Math team · Updated 2026-07-29

Reviewed against published sources. Corrections welcome. Editorial policy · Corrections

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Example problems

Stellar (10 M☉) - Stellar black hole (10 M☉) → Rₛ ~ 29.5 km, Tₕ ~ 6.2 × 10⁻⁹ K
Sgr A* (4×10⁶ M☉) - The horizon of Sgr A* is 0.0790 AU — a fifth of the way from the Sun to Mercury — with the photon sphere at 0.118 AU and the innermost stable orbit at 0.237 AU. The tidal acceleration across a person at the horizon reads 1.16 mm/s², about a ten-thousandth of Earth's gravity: you would cross it without feeling anything.
TON 618 (6.6×10¹⁰ M☉) - A horizon radius of 0.0206 light-years, which light takes seven and a half days to cross, and about 1,300 AU — eight times further out than Voyager 1. The Hawking temperature is 9.35 × 10⁻¹⁹ K and the evaporation time 6.03 × 10⁹⁹ years, so nothing about this object is in a hurry.
Primordial (10⁻⁸ M☉) - A horizon 29.5 μm across, smaller than a grain of sand, and a Hawking temperature of 6.17 K — above the 2.725 K microwave background. That is the line that matters: this one radiates away more than it absorbs and can genuinely evaporate, while every stellar-mass hole is colder than the sky and still growing. Tidal acceleration at the horizon: 1.85 × 10²⁶ m/s².