The Schwarzschild radius calculator above returns the radius at which a given mass would become a black hole — the size of the event horizon, the surface from which light cannot escape. It applies to any mass, not just to astronomical ones, and the results for everyday objects are part of the point: a person's Schwarzschild radius is smaller than a proton by many orders of magnitude, which is why nothing you can pick up will ever collapse under its own weight.
Arb Digital publishes free calculators that give the supporting quantities alongside the headline one, because a single radius in metres is hard to interpret on its own. This page also reports the mean density inside the horizon, the horizon area, the Hawking temperature and the evaporation lifetime, and those four together tell a much more interesting story than the radius does by itself.
What This Schwarzschild Radius Calculator Does
You enter a mass and it returns the Schwarzschild radius, formatted in whichever unit makes it readable — metres for small masses, kilometres for stellar ones, astronomical units for the supermassive case. Presets cover the Sun, the Earth, Jupiter, our own galactic centre black hole, the M87 black hole that was the first ever imaged, and a person, so you can see the scaling across sixty orders of magnitude of mass.
The grid gives the four derived figures. Mean density is the mass divided by the volume of a sphere of that radius, which is the number that shows how counter-intuitive large black holes are. Horizon area matters because black hole entropy is proportional to it rather than to volume. Hawking temperature is the thermal radiation the horizon emits, and evaporation time is how long the object would take to radiate itself away if nothing ever fell in.
The gravitational constant is an editable field because it is genuinely uncertain. Every other constant in these formulas is exact by definition in the SI, but G is measured, and it is measured worse than almost anything else in physics.
How to Use It
- Choose a preset or type a mass. Presets set both the value and the unit. Typing your own value switches the menu to custom so you always know what is being used.
- Pick the mass unit that keeps your number readable. Solar masses for stars and black holes, Earth masses for planets, kilograms for everything else. Scientific notation works in the box.
- Leave the gravitational constant alone unless you have a reason. The default is the current CODATA value. Change it only if you are testing sensitivity to its uncertainty.
- Read the density figure next. It is the fastest way to see whether a given mass is anywhere near collapsing, and it is where the surprising behaviour of large black holes shows up.
- Compare the evaporation time against the age of the universe. For anything of stellar mass it is longer by dozens of orders of magnitude, which is why no astrophysical black hole has ever evaporated.
The Formula: How the Schwarzschild Radius Is Calculated
The Schwarzschild radius is rs = 2GM ÷ c2, where G is the gravitational constant, M is the mass and c is the speed of light. It comes out of the Schwarzschild solution to Einstein's field equations, but the same expression falls out of a much simpler argument: set the Newtonian escape velocity equal to the speed of light and solve for the radius. That the two agree exactly is a coincidence of this particular quantity rather than a general rule.
Work the Sun through it. Using G = 6.6743 × 10−11 m³ kg−1 s−2 from the NIST CODATA value for the Newtonian constant of gravitation, and the nominal solar mass of 1.98847 × 1030 kg, the numerator 2GM is 2.654 × 1020. Dividing by c2 = 8.98755 × 1016 gives 2,953 m. The Sun would have to be compressed to a ball under three kilometres across to become a black hole.
The relationship is exactly linear in mass, which makes the scaling easy to carry in your head: about 3 km per solar mass. A ten-solar-mass stellar black hole has a 30 km horizon. The four-million-solar-mass black hole at our galactic centre has a horizon about 12 million kilometres across, roughly a tenth the size of Mercury's orbit.
The other three figures follow from the radius. Mean density is M ÷ (4πrs3 ÷ 3), horizon area is 4πrs2, Hawking temperature is ħc3 ÷ (8πGMkB), and evaporation time is 5120πG2M3 ÷ (ħc4).
Why Bigger Black Holes Are Less Dense
This is the result that catches people, and it is a direct consequence of the linear scaling. The radius grows in proportion to mass, so the volume inside the horizon grows as mass cubed. Density is mass over volume, so it falls as one over mass squared.
Run the numbers. A solar-mass black hole has a mean density inside its horizon of about 1.8 × 1019 kg/m³, several times denser than an atomic nucleus. Sagittarius A*, at roughly four million solar masses, comes out around 106 kg/m³. Push to the largest known supermassive black holes, tens of billions of solar masses, and the mean density drops below that of water — and for a hypothetical black hole of a hundred billion solar masses, below that of air.
The physical reading is that forming a black hole does not require extreme density in itself, only enough mass inside a small enough radius. At sufficiently large scales that condition is met at densities that would feel entirely unremarkable. It is worth stressing that the mean density is a bookkeeping figure — mass divided by horizon volume — and not a description of any actual material distribution inside, which general relativity says collapses to a singularity regardless.
Hawking Temperature and Why Nothing Has Evaporated
A black hole is not perfectly black. Quantum field theory in curved spacetime predicts that the horizon radiates a thermal spectrum at a temperature inversely proportional to the mass, which means large black holes are cold and small ones are hot.
A solar-mass black hole sits at about 6 × 10−8 K, sixty billionths of a degree above absolute zero. The cosmic microwave background is currently at 2.7 K, so every astrophysical black hole in the universe absorbs vastly more energy from the background than it radiates. None of them are shrinking; all of them are growing, and they will continue to until the expansion of the universe cools the background below their own temperature.
The evaporation time compounds that. It scales as mass cubed, so a solar-mass black hole would take around 2 × 1067 years to evaporate — against a current age of the universe of about 1.4 × 1010 years. Only a black hole of asteroid mass or below would have a lifetime short enough to matter on any timescale we can observe, and none are known to exist.
Everyday Masses and Why Nothing Collapses
The formula does not care whether the mass is astronomical. A 70 kg person has a Schwarzschild radius of about 10−25 metres, which is around ten billion times smaller than a proton. The Earth's is 8.9 millimetres — the whole planet compressed into something you could hold between finger and thumb, if it did not immediately stop being holdable.
That is the honest answer to why ordinary matter never collapses. Gravity at these masses is not remotely strong enough to overcome electromagnetic repulsion between atoms, and the radius at which it would win is far below any scale where our physics has been tested. The gravitational force calculator shows how weak the force actually is between everyday objects, and the escape velocity calculator gives the same boundary from the other direction, since the Schwarzschild radius is precisely where escape velocity reaches the speed of light.
Where This Sits Next to Our Other Relativity Tools
This page computes a length from a mass. It says nothing about what happens to clocks or rulers near that length, which is a separate calculation. Time near a massive body runs slow by a factor involving the ratio of the Schwarzschild radius to your distance from the centre, and the time dilation calculator covers that side of relativity.
For the energy content of the mass itself rather than its gravitational reach, the mass energy equivalence calculator applies the other famous relation. For orbits around a compact object at distances well outside the horizon, where Newtonian mechanics is still a good approximation, the Kepler's third law calculator relates orbital period to separation. And for comparing the mean density figure here with ordinary materials, the density calculator is the everyday counterpart.
NASA's Imagine the Universe introduction to black holes is a good non-mathematical companion to all of this, and it defines the event horizon in exactly the terms used here: the point of no return, where escape velocity equals the speed of light.
Arb Digital builds free tools like this one because useful pages earn attention. If you want tools, calculators or content built for your own audience, we can help.
Browse All Free Tools Talk to Arb DigitalCommon Mistakes to Avoid
- Assuming a black hole must be dense — mean density falls as one over mass squared, so the largest supermassive black holes are less dense than water inside their horizons.
- Reading the Schwarzschild radius as a physical surface — it is a boundary in spacetime, not a material shell. Nothing special happens locally to an infalling object as it crosses.
- Forgetting the factor of two — the radius is 2GM over c squared. Dropping the 2 halves every answer, and it is the most common slip in the algebra.
- Mixing up mass units — one solar mass is 1.98847 × 1030 kg. Entering 1 with kilograms selected asks about a one-kilogram object, and the answer differs by thirty orders of magnitude.
- Expecting Hawking radiation to be observable — every known black hole is far colder than the cosmic microwave background, so all of them are absorbing more than they emit.
Related Free Tools From Arb Digital
Approach the same boundary from the escape velocity side with the escape velocity calculator, and see how weak gravity is at ordinary masses with the gravitational force calculator. For the relativistic behaviour of clocks use the time dilation calculator, and for the energy locked in a mass use the mass energy equivalence calculator. Orbits well outside the horizon are handled by the Kepler's third law calculator, and the density calculator puts the density figures here into everyday terms. Browse everything at the free online tools hub.
Frequently Asked Questions
About 2,953 metres, just under three kilometres. The Sun would have to be compressed from its current 700,000 kilometre radius into a ball less than three kilometres across to become a black hole. The scaling is linear, so it is roughly three kilometres per solar mass.
Because the radius grows in proportion to mass while the volume inside it grows as mass cubed, so mean density falls as one over mass squared. The largest supermassive black holes have a mean density inside the horizon lower than that of water.
Every mass has one in the sense that the formula returns a number, but that number is only physically meaningful if the object is actually compressed inside it. For a person the radius is around ten to the minus twenty-five metres, far smaller than a proton, so nothing about ordinary matter approaches it.
No. It is a boundary in spacetime beyond which no path leads back out. An object falling through a large black hole's horizon notices nothing locally at the moment of crossing. The horizon is defined by where light can go, not by any material there.
Because evaporation time scales as mass cubed and Hawking temperature scales as one over mass. A solar-mass black hole is around sixty billionths of a degree above absolute zero, far colder than the 2.7 kelvin cosmic microwave background, so it absorbs more than it radiates and grows rather than shrinks.
Because it is the least precisely measured of the fundamental constants, with a relative uncertainty of about two parts in a hundred thousand. Every other constant used here is exact by SI definition, so all the uncertainty in these results traces back to that one number.
It gives the same expression, which is a coincidence of this particular quantity rather than a sign that the Newtonian treatment is valid. The correct derivation comes from the Schwarzschild solution to Einstein's field equations. Other relativistic results do not agree with their Newtonian analogues.
This tool is provided for educational and study use. It applies the non-rotating, uncharged Schwarzschild solution, so it does not describe rotating Kerr black holes, whose horizons are smaller than these figures for the same mass.
