Schwarzschild Radius Calculator
Enter the mass of any object to find its Schwarzschild radius, the critical size it would need to be compressed to in order to become a black hole. Choose kilograms, solar masses, or Earth masses, and the calculator returns the event horizon radius in auto-scaled units (mm, km, AU, or light-years), the surface gravitational field at the event horizon, and the average density needed to collapse that mass into a black hole. You can also work backwards: enter a known radius to recover the mass. A chart shows how radius scales with mass across many orders of magnitude.
What is the Schwarzschild radius?
The Schwarzschild radius (symbol rs) is the critical radius below which the gravitational field of an object becomes so strong that light itself cannot escape. Any mass compressed within a sphere of this radius forms a black hole, and the surface of that sphere is the event horizon. The concept comes directly from Karl Schwarzschild's 1916 solution to Einstein's field equations of general relativity, derived just weeks after Einstein published the theory. For an ordinary object like a star or planet, the Schwarzschild radius is far smaller than the object's actual size, so no black hole forms. For a 10-solar-mass object, the Schwarzschild radius is about 30 km, and an object that dense is a stellar black hole.
The formula and physical constants
The Schwarzschild radius formula is rs = 2GM/c², where G = 6.6743 × 10⁻¹¹ N·m²·kg⁻² is Newton's gravitational constant, M is the mass of the object in kilograms, and c = 2.99792458 × 10⁸ m/s is the speed of light. The factor 2 arises because this is the radius at which the Newtonian escape velocity equals exactly c, which also happens to be the general-relativistic event horizon for a non-rotating (Schwarzschild) black hole. The relationship is linear: doubling the mass doubles the Schwarzschild radius. A solar-mass object has rs = 2.95 km, so a 10 solar-mass object has rs = 29.5 km.
Gravitational field at the event horizon
The classical approximation for the gravitational field strength at the event horizon is g = GM/rs² = c⁴/(4GM). This quantity can be enormous for low-mass black holes and surprisingly mild for supermassive ones. A 10-solar-mass stellar black hole has a surface gravity of roughly 1.5 × 10¹² m/s², about 150 billion times Earth's surface gravity. But Sagittarius A*, with 4 million solar masses, has a surface gravity of only about 600 m/s² because the Schwarzschild radius grows faster than the mass. This is also why large black holes are paradoxically less destructive close to their event horizons: the tidal forces across a human-scale distance are tolerable at the event horizon of a supermassive black hole.
Average density needed for collapse
A surprising consequence of the Schwarzschild formula is that, for very massive objects, the average density required to form a black hole is extremely low. The average density inside a black hole's Schwarzschild radius scales as rho = 3c⁶/(32piG³M²), which falls as 1/M². A stellar-mass black hole requires nuclear density (~10¹⁸ kg/m³) to form, but a billion-solar-mass supermassive black hole could form at densities lower than water. This tells us that 'black hole' does not necessarily mean 'extremely dense object', just extremely compact relative to its own mass.
Schwarzschild radius of known objects
| Object | Mass (kg) | Schwarzschild radius | Actual radius | Is a black hole? |
|---|---|---|---|---|
| Moon | 7.34 × 10²² | 0.109 mm | 1 737 km | No |
| Earth | 5.97 × 10²⁴ | 8.87 mm | 6 371 km | No |
| Sun | 1.99 × 10³⁰ | 2.95 km | 696 000 km | No |
| Neutron star (1.4 M☉) | 2.78 × 10³⁰ | 4.13 km | ~10 km | No (near limit) |
| Stellar BH (10 M☉) | 1.99 × 10³¹ | 29.5 km | 29.5 km | Yes |
| Sagittarius A* (4.15 × 10⁶ M☉) | 8.26 × 10³⁶ | 1.22 × 10¹⁰ m (0.08 AU) | 0.08 AU | Yes |
| M87* (6.5 × 10⁹ M☉) | 1.29 × 10⁴⁰ | 1.92 × 10¹³ m (128 AU) | 128 AU | Yes |
Computed from rs = 2GM/c². Objects below their own Schwarzschild radius are black holes.
Frequently asked questions
What is the Schwarzschild radius of the Sun?
The Sun's Schwarzschild radius is approximately 2.95 km (about 1.83 miles). The Sun's actual radius is about 696,000 km, so the Sun would need to be crushed to less than 3 km across to become a black hole. In practice, the Sun will never collapse to that density because it lacks sufficient mass to overcome electron and neutron degeneracy pressure.
What is the Schwarzschild radius of Earth?
Earth's Schwarzschild radius is approximately 8.87 mm, about the size of a small marble. To become a black hole, all of Earth's mass would have to be squeezed into a sphere less than 9 mm in diameter. Earth's actual radius is about 6,371 km, so it is many orders of magnitude larger than its Schwarzschild radius.
Is the Schwarzschild radius the same as the event horizon?
For a non-rotating (Schwarzschild) black hole, yes, they are the same thing. The Schwarzschild radius is the radius of the event horizon: the surface inside which escape velocity exceeds the speed of light. For a rotating (Kerr) black hole, the geometry is more complicated and the event horizon is smaller than the static-limit surface, but the Schwarzschild radius still gives a useful order-of-magnitude estimate for the scale of the system.
Does a large black hole have a larger Schwarzschild radius than a small one?
Yes, the Schwarzschild radius is directly proportional to the mass: rs = 2GM/c². Double the mass and you double the radius. A 10 solar-mass black hole has a Schwarzschild radius of about 29.5 km, and a 20 solar-mass black hole has a Schwarzschild radius of about 59 km. Supermassive black holes have Schwarzschild radii comparable to the size of our Solar System.
Why does the formula have a factor of 2?
The factor of 2 in rs = 2GM/c² emerges from both the Newtonian and relativistic derivations, though for subtly different reasons. In Newtonian gravity, setting escape velocity equal to c gives v = sqrt(2GM/r), so r = 2GM/c² is where that escape velocity equals c exactly. In general relativity, Schwarzschild's exact solution places the event horizon at precisely the same radius. The coincidence is not accidental: Schwarzschild's metric was designed to recover Newtonian gravity in the weak-field limit.
Can I use this calculator to find the mass of a black hole from a known event horizon?
Yes. Switch the "Solve for" input to "Mass (from radius)", enter the known Schwarzschild radius in metres, and the calculator will return the corresponding mass in solar masses and kilograms. This is the direct algebraic inverse: M = rs × c² / (2G).
What is the density needed to form a black hole from Earth's mass?
If you compressed all of Earth's mass (5.97 × 10²⁴ kg) into a sphere of radius 8.87 mm, the average density inside would be about 1.84 × 10³⁰ kg/m³, roughly a hundred trillion times denser than atomic nuclei. That kind of compression is not physically possible with any known process for a body as light as Earth.