Schwarzschild Radius Calculator

Background.

The Schwarzschild radius is the size of the event horizon of a non-rotating, uncharged black hole: r_s = 2GM/c². Put a mass in, and the calculator returns the radius inside which nothing — not even light — can get back out, together with the horizon's area, its mean density, its surface gravity, the photon sphere and the innermost stable circular orbit. Run it the other way and it returns the mass a horizon of a given size must contain.

What makes this formula unusual among the equations on this site is that it is not a fit and not an approximation. Karl Schwarzschild solved Einstein's field equations for a static, spherically symmetric vacuum in 1916, within months of the field equations being published, and Birkhoff's theorem later proved that his solution is the only one of that symmetry. Every quantity on this page — the horizon at 2GM/c², the photon sphere at 1.5 r_s, the innermost stable circular orbit at 3 r_s, the horizon area at exactly 4πr_s² — is an exact consequence of that geometry, with no empirical calibration anywhere. The only uncertainty in your answer is the uncertainty in the constants you feed it, and that is dominated entirely by G, whose CODATA 2022 relative standard uncertainty of 2.2 × 10⁻⁵ makes it the least precisely measured constant in this calculation by a factor of millions.

Read the answer with three conventions in mind, because each one changes what the number means. First, r_s is an areal coordinate radius. It is defined so that the horizon's proper area comes out to exactly 4πr_s², not so that it measures a distance you could walk. There is no meaningful "distance from the centre to the horizon" in Schwarzschild geometry, because inside the horizon the radial direction is timelike. Second, the solution assumes zero spin and zero charge. Real astrophysical black holes spin, sometimes close to the theoretical maximum, and a maximally rotating Kerr black hole of the same mass has its outer horizon at GM/c² — exactly half the Schwarzschild answer. Charge is negligible in practice, since ambient plasma neutralises any net charge almost immediately, but spin is not. Treat the Schwarzschild radius as the correct answer for a non-spinning hole and an upper bound for a spinning one of the same mass. Third, the mean density this page reports divides the mass by the volume of a Euclidean sphere of radius r_s. Space inside a horizon is not Euclidean, so that figure is a scaling aid rather than a measurable density — but the scaling it reveals is genuine and startling, because r_s grows linearly with mass while the volume grows as the cube, so density falls as 1/M².

That 1/M² scaling is the most counter-intuitive result the calculator produces, and it is worth stating up front rather than burying it. A one-solar-mass black hole has a mean horizon density of 1.8 × 10¹⁹ kg/m³, roughly a hundred times nuclear matter. Sagittarius A*, at 4.297 million solar masses, comes out near 10⁶ kg/m³ — about a thousand times water. M87*, at 6.5 billion solar masses, comes out at 0.44 kg/m³, roughly a third the density of air at sea level. Nothing about the matter changes; only the ratio of a linear radius to a cubic volume does.

The calculator is deliberately narrow. It answers the geometry question and nothing else. It does not compute the Hawking temperature or evaporation time — that is a separate page — and it does not model accretion, jets, tidal disruption or gravitational-wave emission. It also will not tell you whether a given mass actually is a black hole. Below about 3.2 solar masses, the Rhoades–Ruffini bound, no known stellar-collapse channel produces one, so for small masses the calculator reports the radius the object would have if it were a black hole, and says so in the result. A 70 kg person has a Schwarzschild radius of 1.04 × 10⁻²⁵ m, about a ten-billionth of a proton's radius; that is a real number and a completely hypothetical object.

Every constant carries a named source and revision. G = 6.67430 × 10⁻¹¹ m³ kg⁻¹ s⁻² and c = 299 792 458 m/s exactly come from CODATA 2022. The solar mass, 1.98841 × 10³⁰ kg, comes from the Particle Data Group's astrophysical constants table, which itself derives it from the IAU 2015 Resolution B3 nominal solar mass parameter GM☉ = 1.3271244 × 10²⁰ m³ s⁻² divided by G. That chain matters, because the well-measured quantity in astronomy is the product GM, not the mass — and since r_s = 2GM/c², the Sun's Schwarzschild radius can be computed without ever knowing G at all.

What is schwarzschild radius calculator?

The Schwarzschild radius of a mass M is r_s = 2GM/c². For a black hole it is the radius of the event horizon: the surface from which no signal, including light, can escape to infinity. For any other object it is a hypothetical radius — the size that mass would have to be compressed to before it became a black hole.

The name comes from Karl Schwarzschild, who in 1916 found the exact solution to Einstein's field equations for the vacuum outside a static, spherically symmetric mass. In Schwarzschild coordinates the metric contains the factor (1 − r_s/r), which vanishes at r = r_s. That vanishing is a coordinate artefact rather than a physical singularity — spacetime is perfectly regular at the horizon, as Eddington and Finkelstein later showed with better coordinates — but the horizon itself is real and observer-independent. What happens there is that the radial direction changes character: inside r_s, moving to smaller r becomes as unavoidable as moving forward in time.

Three other radii follow from the same metric and are reported here because they are what observations actually see. The photon sphere at 1.5 r_s is where light can orbit in an unstable circular path; it is responsible for the bright ring in black-hole images. The innermost stable circular orbit (ISCO) at 3 r_s is the closest a massive particle can orbit without spiralling in, and it sets the inner edge of a thin accretion disc and therefore the maximum efficiency of accretion. The Killing surface gravity κ = c⁴/(4GM) is the horizon's gravitational strength as measured from infinity, and it is the quantity that appears in the Hawking temperature and in the first law of black-hole mechanics.

What the Schwarzschild radius is not: it is not the radius of a physical object, it is not a distance you could measure with a ruler, and it is not the size of a spinning black hole. For a rotating (Kerr) hole the outer horizon lies between GM/c² and 2GM/c², reaching the lower value at maximal spin.

How to use this calculator.

1. 01Choose whether you are going from a mass to a horizon radius, or from a horizon radius back to a mass.
2. 02Enter the mass in kilograms. The field's hint lists reference masses for the Sun, Earth, Jupiter, Sagittarius A* and M87* so you do not have to look them up.
3. 03Or, in the radius route, enter an event-horizon radius in kilometres — this is what you do when a paper quotes a horizon size and you want the mass behind it.
4. 04Read the Schwarzschild radius in kilometres, metres, solar radii and astronomical units. The last two only become useful for supermassive holes.
5. 05Look at the photon sphere (1.5 r_s) and ISCO (3 r_s) if you care about what an accretion disc or an imaging campaign would see.
6. 06Read the conventions note before quoting anything. It states the zero-spin assumption — a maximally spinning hole of the same mass has a horizon half this size — and flags masses below the neutron-star bound, where the answer is hypothetical.

The formula.

The derivation is short and, unusually, exact. Schwarzschild's 1916 vacuum solution for a static, spherically symmetric mass has a metric coefficient (1 − 2GM/(rc²)). That factor vanishes at r = 2GM/c², and that radius is the event horizon. Birkhoff's theorem guarantees the solution is unique for this symmetry, so there is no modelling choice to make and no parameter to fit.

A useful accident of history: the same expression falls out of Newtonian mechanics if you set the escape velocity √(2GM/r) equal to c and solve for r. John Michell did exactly that in 1783 and predicted "dark stars". The agreement is a coincidence of algebra, not of physics — the Newtonian argument treats light as a corpuscle slowing under gravity, which is wrong, and it gets the photon sphere, the ISCO and the causal structure of the horizon completely wrong. But it is why an escape-velocity calculator and this one land on the same number.

The other three radii come from the same metric. Circular photon orbits require r = 3GM/c² = 1.5 r_s. The innermost stable circular orbit for a massive particle sits at r = 6GM/c² = 3 r_s, below which the effective potential has no minimum. The Killing surface gravity is κ = c⁴/(4GM), which satisfies the tidy identity κ · r_s = c²/2 — a relation the tests assert, because it is a dimensional check that cannot pass by accident.

Units and dimensions. Mass is in kilograms and horizon radius in kilometres on input; internally everything is SI. G carries m³ kg⁻¹ s⁻², so GM/c² has dimensions of (m³ s⁻²)/(m² s⁻²) = metres, which is the dimensional-analysis guard the test suite runs. The horizon area is in m², the mean density in kg m⁻³, and the surface gravity in m s⁻². Magnitudes here span an absurd range — from 10⁻²⁵ m for a person to 10¹³ m for M87* — which is why the outputs are rounded to twelve significant digits rather than to a fixed number of decimal places.

Rounding stage. There is no intermediate rounding anywhere. All arithmetic runs at forty significant digits in a high-precision decimal library, and rounding happens exactly once, at the moment the result is returned. Decimal-place rounding was deliberately rejected: a 70 kg body's horizon radius of 1.04 × 10⁻²⁵ m would collapse to a flat zero under ten-decimal-place rounding, which is a wrong answer disguised as a rounded one, so decade-spanning quantities use twelve significant digits instead.

Significant figures in practice. Do not quote more than five significant digits of any result as physically meaningful. G is known to only 2.2 × 10⁻⁵ relative uncertainty (CODATA 2022), so a horizon radius computed from a mass in kilograms inherits about one part in 45 000 of uncertainty no matter how precise the mass is. The exception is a body whose GM product is measured directly, which is the case for every object in the solar system: for the Sun, r_s = 2GM☉/c² can be evaluated from the IAU 2015 B3 nominal GM☉ = 1.3271244 × 10²⁰ m³ s⁻² without G appearing at all, giving 2953.250076 m to nine digits.

Invalid-domain behaviour. Zero or negative mass raises a field error rather than returning zero or a negative radius: a zero-mass horizon has zero area and an undefined density and surface gravity, and negative mass is not a physical solution of the field equations. Zero or negative horizon radius raises the same kind of error in the inverse route. Below 3.2 solar masses the calculator still returns numbers but attaches a note, because the Rhoades–Ruffini (1974) absolute upper bound on neutron-star mass means no known formation channel produces a black hole that light — the figure is then the radius the mass would have if it were a black hole, not evidence that it is one.

A worked example.

Frequently asked questions.

References& sources.

1. [1]Schwarzschild, K. (1916), "Über das Gravitationsfeld eines Massenpunktes nach der Einsteinschen Theorie", Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften zu Berlin, 189–196. The original exact solution from which r_s = 2GM/c² follows. Print/bibliographic reference; an English translation is widely reprinted (arXiv:physics/9905030). Not independently re-fetched — cited as the source of the solution, not as the source of any number used here.
2. [2]Particle Data Group (Workman et al.), Review of Particle Physics, "Astrophysical Constants and Parameters", Table 2.1, revised August 2023 by D. E. Groom and D. Scott; 2024 edition PDF retrieved 2026-07-29. Prints: Newtonian constant of gravitation G_N = 6.67430(15) × 10⁻¹¹ m³ kg⁻¹ s⁻²; Schwarzschild radius of the Sun 2G_N M☉/c² = 2.9532501 km; solar mass M☉ = 1.98841(4) × 10³⁰ kg; Schwarzschild radius of the Earth 2G_N M⊕/c² = 8.870056 mm; Earth mass M⊕ = 5.97217(13) × 10²⁴ kg; astronomical unit 149 597 870 700 m exact. Independent of the IAU documents; open access. This is the second, independent authority used to verify the engine.
3. [3]NIST/CODATA 2022, "Newtonian constant of gravitation": G = 6.674 30 × 10⁻¹¹ m³ kg⁻¹ s⁻², standard uncertainty 0.000 15 × 10⁻¹¹, relative standard uncertainty 2.2 × 10⁻⁵. Retrieved 2026-07-29. This is the dominant uncertainty in every result on this page. Standards body; open access.
4. [4]NIST/CODATA 2022, "Speed of light in vacuum": c = 299 792 458 m s⁻¹, exact (it defines the metre). Retrieved 2026-07-29. Standards body; open access.
5. [5]Prša, A., et al. (2016), "Nominal Values for Selected Solar and Planetary Quantities: IAU 2015 Resolution B3", Astronomical Journal 152, 41; arXiv:1510.07674, full text retrieved 2026-07-29. Table 1 gives the nominal solar mass parameter (GM)☉ = 1.3271244 × 10²⁰ m³ s⁻², the nominal solar radius 6.957 × 10⁸ m, and the nominal terrestrial mass parameter (GM)⊕ = 3.986004 × 10¹⁴ m³ s⁻². The resolution states that if SI masses are needed they should be expressed as (GM)/G with the chosen G declared — which is exactly the chain used here. Peer-reviewed; open-access preprint.
6. [6]Rhoades, C. E. and Ruffini, R. (1974), "Maximum Mass of a Neutron Star", Physical Review Letters 32, 324. Derives an absolute upper bound of about 3.2 M☉ on the mass of a non-rotating neutron star from general relativity, causality and minimal equation-of-state assumptions. Used here only to decide when to flag a result as hypothetical. Peer-reviewed; paywalled at APS (abstract and citation freely visible).
7. [7]Event Horizon Telescope Collaboration (2019), "First M87 Event Horizon Telescope Results. I. The Shadow of the Supermassive Black Hole", Astrophysical Journal Letters 875, L1. Source of the M87* mass of 6.5 × 10⁹ M☉ quoted in the FAQs and of the observational relevance of the photon sphere. Peer-reviewed; open access.
8. [8]Michell, J. (1784), "On the Means of Discovering the Distance, Magnitude, etc. of the Fixed Stars...", Philosophical Transactions of the Royal Society of London 74, 35–57. The 1783 Newtonian "dark star" argument that coincidentally reproduces 2GM/c². Historical/bibliographic reference; open access via JSTOR and the Royal Society archive.