This Schwarzschild radius calculator finds the size of the event horizon for any mass — Rs = 2GM/c² — the radius to which you’d have to crush an object before it becomes a black hole.

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The Crush-To Size of Everything

Three months after Einstein published general relativity in 1915, Karl Schwarzschild — solving the field equations from a WWI trench — produced the first exact solution, and inside it sat this radius: Rs = 2GM/c². Squeeze a mass within its Schwarzschild radius and the escape velocity at its surface reaches the speed of light; space-time seals it off behind an event horizon, and a black hole is the result. The numbers recalibrate your intuition about density: the Sun’s Rs is 2.95 km (its actual radius is 696,000 km — it would need to shrink by a factor of ~236,000), Earth’s is 8.87 mm — the whole planet inside a marble — and a person-sized mass corresponds to ~10⁻²⁵ m, far below atomic scales, which is why only astrophysical masses (or the early universe’s extremes) make black holes in practice. The linear scaling is the memorable part: ~3 km per solar mass, so the formula becomes mental arithmetic — a 10-solar-mass stellar black hole spans ~30 km, a city; a billion-solar-mass quasar engine spans ~3 billion km, a solar system. Michell and Laplace had guessed the same formula in the 1780s from Newtonian escape velocity alone — a famous coincidence, since the GR derivation is entirely different and the agreement is luck of the factor 2.

Two clarifications keep the concept honest. First, Rs is not a physical surface — it’s a coordinate boundary (the “surface” of a non-rotating black hole is the horizon itself, a one-way membrane in the geometry, not a material shell), and an infalling observer notices nothing special crossing a large hole’s horizon; the tidal violence scales inversely with mass, which is why supermassive horizons are gentle and stellar ones shred. Second, real astrophysical black holes rotate, and rotation (Kerr geometry) shrinks the horizon — down to Rs/2 at maximum spin — and adds an ergosphere; the Schwarzschild value remains the reference everyone quotes first. The formula also anchors the famous density paradox: average density inside Rs falls as 1/M², so a billion-solar-mass hole has the “density” of air at its horizon scale — black holes need not be dense, merely compact relative to their own (mass-proportional) radius. Enter any mass above — the calculator takes solar masses, the astronomer’s unit — and get the horizon size that a century of relativity still computes with Schwarzschild’s trench mathematics.

Worked Examples (Verified)

  • The Sun: 1 solar mass → Rs = 2.954 km.
  • Earth: 3.0×10⁻⁶ solar masses → Rs = 8.87 mm.
  • Sagittarius A*: 4.3×10⁶ solar masses → Rs ≈ 12.7 million km (≈ 0.085 AU).
  • A stellar black hole: 10 solar masses → 29.5 km — the size of a city.

Schwarzschild Radius Discipline

  • Rs is a radius, not a diameter — halve your intuition accordingly; the horizon’s diameter of the Sun’s hole is ~5.9 km.
  • Rotation shrinks the real horizon: an extremal Kerr hole’s horizon is Rs/2, and frame-dragging adds an ergosphere outside it.
  • Nothing special happens LOCALLY at a big horizon — tides scale as 1/M²; the “spaghettification at the horizon” image belongs to small holes.
  • The Newtonian derivation (set escape velocity = c) lands on the same formula by coincidence of factors — quoting it as the reason is a textbook shortcut, not the physics.
  • Density inside Rs is a volume-average fiction that falls as 1/M² — useful for wonder, meaningless as a material property.

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Frequently Asked Questions

How do you calculate the Schwarzschild radius?

Rs = 2GM/c², with G = 6.674×10⁻¹¹, M the mass in kilograms and c = 2.998×10⁸ m/s. Because everything except M is constant, Rs is strictly proportional to mass: about 2.95 km per solar mass. This calculator takes mass in solar units (1 = 1.989×10³⁰ kg) and returns the radius in the most readable unit. The Sun: 2.95 km. Earth (3×10⁻⁶ solar masses): 8.87 millimetres.

What happens if an object is compressed within its Schwarzschild radius?

Collapse becomes unstoppable. Inside Rs, general relativity shows all future paths point inward — the roles of space and time effectively swap — so no pressure, force or signal can maintain the surface; the mass continues to a singularity and the Schwarzschild surface becomes an event horizon, a one-way boundary signals cannot cross outward. This is not hypothetical machinery: stellar cores collapsing past their Rs is precisely how stellar-mass black holes form in supernovae.

Is the Schwarzschild radius the size of the black hole?

It is the size of the event horizon of a NON-rotating black hole — the operational “size” astronomers quote. The mass itself, in classical GR, is concentrated at the central singularity (a point for Schwarzschild, a ring for rotating Kerr holes). Real black holes spin, and their horizons are smaller — as low as half the Schwarzschild radius for a maximally spinning hole — plus an ergosphere bulging beyond. When headlines give a black hole’s size, Schwarzschild geometry is the default yardstick behind it.

What is the Schwarzschild radius of Earth or a person?

Earth: 8.87 mm — compress the planet into a marble and it would be a black hole (holding it there is another matter entirely; nothing is doing that compressing). A 70 kg person: about 1.0×10⁻²⁵ m — a hundred billion times smaller than an atom. The exercise’s point is the scaling: everyday masses have absurdly small horizons, which is why black holes in nature come from collapsed stars and galactic centers, not from anything on a human scale.

Why does the Newtonian escape-velocity formula give the same answer?

Set √(2GM/R) = c and solve: R = 2GM/c² — identical. Michell (1783) and Laplace noticed this “dark star” consequence two centuries before relativity. Physicists regard the match as a genuine coincidence: the Newtonian picture (light slowed and falling back like a projectile) is wrong in mechanism — in GR, light at the horizon isn’t slowed, space-time itself is structured so all paths lead inward — and several factors of 2 that could have differed happened not to. It’s a memorable shortcut and a false derivation; both facts are worth knowing.

How dense is a black hole?

Depends on mass — counterintuitively, bigger means fluffier. “Density” computed as M divided by the horizon’s volume scales as 1/M²: a 10-solar-mass hole averages ~2×10¹⁷ kg/m³ (nuclear density), the Sun-as-a-hole ~1.8×10¹⁹, but a billion-solar-mass hole averages only ~19 kg/m³ — less than water, near the density of air at altitude — and the largest known holes average below the density of air at sea level. This is why supermassive horizons can be crossed gently: compactness, not density, is what makes a black hole.

What is Sagittarius A*’s Schwarzschild radius?

About 12.7 million km (0.085 AU) for its ~4.3 million solar masses — a horizon radius about 18 times the Sun’s radius (diameter ~25 million km), sitting 26,000 light-years away. The Event Horizon Telescope’s famous 2022 image shows the glowing ring of hot gas around this shadow; the dark region spans about 5.2 Rs across, because light-bending enlarges the photon-capture shadow to a radius of √27/2 × Rs ≈ 2.6 Rs — the enlargement the calculator’s simple Rs doesn’t include.

Is this Schwarzschild radius calculator free?

Yes — free, instant, no sign-up. Enter a mass in solar masses for its Schwarzschild radius in the clearest unit.