Schwarzschild Radius Calculator
Enter any mass in kg, tonnes, Earth masses or Solar masses to find its Schwarzschild radius — the event-horizon size the object would have as a black hole, with interpretation of the result.
Find the Event-Horizon Radius of Any Mass
Schwarzschild Radius Formula
Where:
- \(r_s\) — Schwarzschild radius, the radius of the event horizon (m)
- \(G\) — gravitational constant, \(6.674 \times 10^{-11}\ \mathrm{m^3\,kg^{-1}\,s^{-2}}\)
- \(M\) — mass of the object (kg)
- \(c\) — speed of light in vacuum, \(299{,}792{,}458\ \mathrm{m/s}\)
The formula gives the critical radius at which the escape velocity of a mass equals the speed of light. Squeeze the whole mass inside this radius and nothing — not even light — can escape: the object becomes a black hole.
Schwarzschild Radius – Calculation Example
Example: a star with a mass equal to the Sun collapses after exhausting its nuclear fuel. What is the event-horizon radius of the resulting black hole? Take \(M = 1\) Solar mass \(= 1.98847 \times 10^{30}\) kg.
\[ r_s = \frac{2 \times 6.674 \times 10^{-11} \times 1.98847 \times 10^{30}}{(2.998 \times 10^{8})^2} = \frac{2.654 \times 10^{20}}{8.988 \times 10^{16}} \approx 2953\ \mathrm{m} \]
Result: the Schwarzschild radius is about 2,953 m, or roughly 2.95 km. All of the Sun's mass compressed within a sphere just under 3 km across would form a black hole with this horizon.
How to Use the Calculator
- Enter the Mass of the object (M) in the number field — for example, 1 for a Sun-mass object.
- Choose the matching Unit from the dropdown: kg, g, tonnes, lb, Earth masses or Solar masses.
- Read the Schwarzschild radius (rs) result, shown in metres and kilometres and converted into cm, mm, miles, feet and astronomical units.
- Check the Details and interpretation block for the escape velocity at the horizon, the average density inside it, and a plain-language explanation of what the number means.
What Is the Schwarzschild Radius?
The Schwarzschild radius is the distance from the centre of a mass at which its escape velocity equals the speed of light. It is named after Karl Schwarzschild, who found this solution to Einstein's general relativity equations in 1916, just months after the theory was published. The sphere of radius \(r_s\) around a collapsed object is its event horizon: the boundary beyond which no signal, matter or light can ever return to the outside universe.
Every object has a Schwarzschild radius — it is simply proportional to its mass. The catch is that for ordinary objects the radius is absurdly small. A 70 kg human has \(r_s \approx 10^{-25}\) m, many orders of magnitude smaller than a proton, which is why people do not spontaneously become black holes. Only when matter is compressed within its own \(r_s\) does gravitational collapse become unstoppable.
Because \(r_s\) scales linearly with mass while volume grows with the cube of the radius, larger black holes are actually less dense on average. A supermassive black hole of a billion Solar masses has an average density comparable to air, while a stellar-mass black hole packs densities far above nuclear matter. The calculator's density output lets you see this counter-intuitive trend directly.
The result is also a handy yardstick in astronomy: a black hole of one Solar mass has \(r_s \approx 2.95\) km, Earth's would be about 8.87 mm, and Sagittarius A*, the Milky Way's central black hole of roughly 4 million Solar masses, has a horizon about 12,000 km across — a sphere a bit larger than Earth.
Assumptions, Limitations and Tips
- Non-rotating, uncharged object. The formula describes an idealized Schwarzschild black hole. Real black holes spin (Kerr black holes) and their horizons are somewhat smaller — up to a factor of 2 for a maximally rotating one. For everyday estimates the difference is usually acceptable.
- Classical treatment. Quantum-gravity effects are ignored. For extremely tiny masses the classical horizon size can be smaller than the Planck length \(\approx 1.6 \times 10^{-35}\) m, where the formula loses physical meaning.
- All mass counted at the centre. The formula assumes the entire mass lies within \(r_s\). A star or planet of the same mass that is larger than its Schwarzschild radius is simply an ordinary object — it only becomes a black hole if compressed inside that radius.
- Use big units for big objects. Entering a huge mass in kilograms may overflow the calculation. Switch the unit dropdown to Solar masses or Earth masses to keep the numbers manageable.
- Working with radiation from hot accretion disks? The disk glows like a blackbody, so the Stefan–Boltzmann law calculator and Wien's displacement law calculator can help you estimate its power output and peak wavelength.
Reference Values
| Object | Approximate mass | Schwarzschild radius |
|---|---|---|
| Human (70 kg) | 70 kg | \(\sim 10^{-25}\) m |
| Earth | \(5.97 \times 10^{24}\) kg | 8.87 mm |
| Sun | \(1.99 \times 10^{30}\) kg | 2.95 km |
| Sagittarius A* (Milky Way centre) | \(\sim 4 \times 10^6\) Solar masses | \(\sim 1.2 \times 10^7\) km |
| M87* (first imaged black hole) | \(\sim 6.5 \times 10^9\) Solar masses | \(\sim 1.9 \times 10^{10}\) km |
You can verify each row by entering the mass in the calculator with the appropriate unit selected — the results above use the same formula and constants.
Frequently Asked Questions
How do I calculate the Schwarzschild radius by hand?
Multiply the mass by 2 and by the gravitational constant, then divide by the speed of light squared: \(r_s = 2GM/c^2\). A handy shortcut: \(r_s \approx 1.48 \times 10^{-27}\) m per kilogram, so just multiply the mass in kg by that factor. For example, 1000 kg gives \(r_s \approx 1.5 \times 10^{-24}\) m — you don't need to handle the huge constants separately.
What is a typical or 'normal' Schwarzschild radius?
It depends entirely on the mass: the radius is about 2.95 km per Solar mass. A black hole of 10 Solar masses (a typical collapsed star) has a horizon near 30 km, Earth's mass would give about 8.87 mm, and supermassive black holes run from millions to billions of km. There is no single 'normal' value — just the linear scaling with mass.
Is the Schwarzschild radius the size of the black hole?
It is the radius of the event horizon, which is the closest thing a black hole has to a 'surface'. Strictly speaking nothing physical exists at the horizon — it is a boundary in spacetime. The singularity lies at the centre, and the matter that formed the hole has collapsed into it, so the horizon encloses mostly empty space plus whatever is falling in.
How do I convert the result to miles, feet or astronomical units?
The calculator does this automatically. If you are computing manually: divide metres by 1609.344 for miles, by 0.3048 for feet, or by \(1.496 \times 10^{11}\) for astronomical units. For example, the Sun's 2953 m horizon is about 1.84 miles, 9690 feet, or \(2 \times 10^{-8}\) AU — smaller than an atom relative to the Earth–Sun distance.
Does my body or everyday objects have a Schwarzschild radius?
Yes, every mass has one — a 70 kg person's is about \(10^{-25}\) m, roughly ten orders of magnitude smaller than a proton. This is completely harmless: the object only becomes a black hole if something compresses it inside that radius, and no physical process can do this to ordinary matter. The number is just a theoretical limit.
Why do bigger black holes have lower average density?
Because mass grows linearly with the Schwarzschild radius while the enclosed volume grows with the cube of the radius: \(\rho = 3c^6/(32\pi G^3 M^2)\). Doubling the mass doubles the radius but eight-times the volume, so average density drops by a factor of four. That is why a billion-Solar-mass black hole averages roughly the density of air, while a stellar-mass one exceeds nuclear densities.
What are common mistakes when using the formula?
The most frequent error is mixing units — always convert the mass to kilograms and use SI values for \(G\) and \(c\) before plugging in. Others include forgetting that the result is in metres (not km), assuming the formula applies to a star's current size (it only matters once the mass fits inside \(r_s\)), and ignoring that rotation shrinks the real horizon by up to a factor of 2 for a Kerr black hole.