Schwarzschild Radius Calculator

Mass of the object (M)

Schwarzschild radius (r)

Surface gravity at the horizon (g)

Enter any one of the three: the mass, the Schwarzschild radius, or the surface gravity. A non-rotating black hole is fixed by a single number, so the calculator fills in the other two, the size of the event horizon, and its density.

Show the event horizon's size and shape

Adds the circumference, surface area, and volume of the event-horizon sphere alongside the radius.

Compare with a real object's size

Enter an object's actual radius and see how many times you would have to compress it for its mass to collapse inside its Schwarzschild radius.

Show the Hawking temperature

A black hole is not perfectly black: it radiates at the Hawking temperature, which is tiny for a stellar hole and rises as the hole loses mass.

Please input values into the fields

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A black hole is a celestial object in which a large mass has been compressed into an extremely small space such that not even light can escape. The boundary of this enclosed region is called the event horizon. For a non-rotating black hole, its radius takes on a special name: the Schwarzschild radius. This calculator relates three quantities that describe a black hole: its mass, the radius of its event horizon, and surface gravity. If any one of these quantities are entered, the other two can be calculated.

If you enter the mass in solar masses, it will calculate the size of the event horizon. Conversely, if you enter a radius, it will tell you what the corresponding black hole mass would be for that size. This tool also calculates the density within the event horizon, geometric properties of a sphere and, where applicable, how much matter is required to compress an actual object down to collapse into one.

What is a Schwarzschild Radius?

The Schwarzschild radius is named after Karl Schwarzschild who in 1916 found the first exact solution to Einstein's field equations. It describes the event horizon of a non-rotating black hole and is the distance from the center at which the escape velocity becomes equal to the speed of light. Objects closer to the center would require an escape velocity that is faster than the speed of light, meaning nothing can escape.

For a spherical object the escape velocity is determined only by the mass, and therefore the event horizon of a Schwarzschild black hole is also determined only by the mass. This explains why this calculator is so simple to use: entering a single number determines the state of the entire black hole.

Equation for black holes.

The Schwarzschild radius can be derived by setting the escape velocity equal to the speed of light.

r_s = \dfrac{2\,G\,M}{c^{2}}

Surface gravity is the strength of the gravitational field at the event horizon. It is calculated by applying Newton's law of universal gravitation to this radius.

g = \dfrac{G\,M}{r_s^{2}} = \dfrac{c^{2}}{2\,r_s}

Each symbol has a unique physical meaning.

Symbol	Meaning	Value or unit
$r_s$	Schwarzschild radius (event horizon)	metres
M	Mass of the black hole	kilograms
g	Surface gravity at the horizon	$\mathrm{m/s^2}$
G	Gravitational constant	$6.674\times10^{-11}\ \mathrm{N\,m^2/kg^2}$
c	Speed of light in vacuum	$2.998\times10^{8}\ \mathrm{m/s}$

How to use this calculator:

Enter one of the three values and select its corresponding unit. Once you have finished entering data, the two remaining values will be shown directly next to the fields, with the radius being presented as the main result. It is not necessary to select the desired value. The empty fields are the magnitudes for which the calculator performs calculations.

You usually input a mass in solar masses to find the radius of the event horizon. For calculations in reverse direction you leave the field for mass blank and instead enter either the radius or surface gravity. By opening an optional section, geometric quantities of the horizon, comparisons with real objects sizes or Hawking temperature can be added.

Example - A black hole with ten times the mass of our Sun

Let's consider a black hole that has ten times the mass of our Sun. Since one solar mass is about 1.989 x 1030 kilograms, this black hole would have a mass of 1.989 x 1031 kilograms. Plugging these values into the formula gives:

r_s = \dfrac{2\,(6.674\times10^{-11})(1.989\times10^{31})}{(2.998\times10^{8})^{2}} \approx 2.95\times10^{4}\ \text{m}

Thus the radius of the event horizon for a black hole with ten times the mass of our Sun would be about 29.5 km, which is roughly the size of a city. Even if the Sun were to somehow collapse into a black hole, its Schwarzschild radius would only be about three kilometers.

How big are black holes?

The Schwarzschild radius is strictly proportional to the mass, so the size of a black hole increases very regularly with increasing mass. A few calculation examples are shown below to illustrate the enormous range.

Object	Mass	Schwarzschild radius
The Earth	$5.97\times10^{24}\ \mathrm{kg}$	about 8.9 mm
The Sun	$1.99\times10^{30}\ \mathrm{kg}$	about 2.95 km
A 10-solar-mass black hole	$1.99\times10^{31}\ \mathrm{kg}$	about 29.5 km
Sagittarius A* (our galaxy's core)	$4.3\times10^{6}\ M_\odot$	about 12.7 million km

The values for Earth are particularly astonishing. To form a black hole, the entire Earth would have to be compressed into a sphere less than one centimeter across, making it over seven hundred million times smaller. This is why normal objects never collapse. Only the enormous pressure of dying massive stars can squeeze matter down to their respective Schwarzschild radii.

The Density Paradox

When you compress the mass of several suns into an area about the size of a city, the average density becomes astonishingly high, exceeding more than ten to the power of eighteen kilograms per cubic meter -- much higher than the density of atomic nuclei. So one tends to think that all black holes have incredibly high densities.

For supermassive black holes the opposite is true. As the radius increases with mass and volume increases as the cube of the radius, average density decreases proportionally to the square of the mass. For a supermassive black hole with a mass of several billion Suns, the average density would be less than that of air. A sufficiently massive black hole could even have a lower density than the best vacuum in a laboratory. The calculator shows this density and allows you to see the effects of entering different masses.

Hawking radiation and black hole temperature

A black hole is not completely dark. Due to quantum effects at the event horizon it emits a weak thermal radiation called Hawking radiation. Its temperature is inversely proportional to its mass.

T = \dfrac{\hbar\,c^{3}}{8\pi\,G\,M\,k_B}

For a black hole with the same mass as our Sun, this temperature is about sixty billionths of a Kelvin, which is so low that it's not measurable and completely overwhelmed by the heat of the cosmic microwave background. Only very small black holes have high temperatures. A black hole gets hotter and radiates faster when it loses mass, eventually leading to an uncontrollable explosion. You can check the temperature of a black hole by opening up the Hawking layer.

Regions where the Schwarzschild radius plays a role.

This concept is central to modern astrophysics. When massive stars collapse they form stellar black holes. Their Schwarzschild radii range from a few kilometers up to several tens of kilometers and determine the size scale of mergers between two black holes that are currently being recorded by gravitational wave detectors.

Supermassive black holes are located at the center of galaxies. Because of their event horizons that span tens of millions of kilometers, the Event Horizon Telescope was able to image the dark silhouettes of black holes in M87 and at the center of our own galaxy.

Frequently asked questions

How do you calculate the Schwarzschild Radius?

You multiply the mass by the gravitational constant G, divide the result by the square of the speed of light and then multiply it again by two. In the SI system of units you get a radius in meters from a mass in kilograms. For a mass of ten solar masses, the calculated radius is about 29.5 kilometers.

What is Earth's Schwarzschild Radius?

It is about 8.9 millimeters. For the Earth to become a black hole, it would have to be compressed without any mass loss into a ball smaller than a marble, and thus more than seven hundred million times smaller than its actual size. This is why everyday objects do not collapse into black holes.

Can a black hole be smaller than its own Schwarzschild radius?

The Schwarzschild radius marks the event horizon, or boundary from which nothing can escape. It is believed that matter itself collapses within a much smaller region so physical reality is less than the horizon. The size of the horizon corresponds to what we perceive as the surface area of the black hole.

Does the schwarzschild radius depend on factors not related to mass?

For a non-spinning and uncharged black hole this is not the case. The radius of the event horizon is completely determined by the mass and is directly proportional to it. When spin and charge are present the situation changes, leading to more general Kerr or Reissner-Nordström solutions, but the Schwarzschild radius is a simple case where it depends only on the mass.

Why are supermassive black holes less dense than water?

Because the radius is proportional to the mass, the volume within the event horizon increases in proportion to the cube of the mass, while the mass itself only increases linearly. Thus the average density decreases as the square of the mass. While the density of a stellar black hole is greater than that of an atomic nucleus, the average density of a supermassive black hole at the center of a galaxy with a mass corresponding to one billion Suns would be less than that of air.

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Disclaimer: This calculator is provided for general informational and educational purposes only. Our calculators are under active development, and results may be inaccurate, incomplete, or unsuitable for your situation. Always verify the figures independently and seek advice from a qualified professional before relying on them. We make no warranties and accept no liability for any loss or decision arising from use of this tool.

References

Wikipedia: Schwarzschild radius
Definition, derivation, and worked values for the Schwarzschild radius.
HyperPhysics: Black Holes
The event horizon, Schwarzschild radius, and black hole basics.
NASA: Black Holes
Overview of stellar and supermassive black holes and their event horizons.