How is the Schwarzschild Radius Calculated?

You multiply the gravitational constant G by two, then you multiply it by the mass and divide that result by the square of the speed of light. The event horizon diameter for a black hole with one solar mass is about 2.95 km, where the radius is proportional to the mass.

What is Earth's Schwarzschild Radius?

It is about 8.9 millimeters in size. A black hole only forms when the whole Earth is compressed to the size of a marble.

Is a black hole itself smaller than its Schwarzschild radius?

Yes. The schwarzschild radius is the event horizon or border surrounding a black hole. It's assumed that the mass itself collapses into a much smaller region.

Why do larger black holes have lower density?

As the radius increases with mass, the volume increases in proportion to the cube of the mass while the increase in mass is linear. Density is defined as mass divided by volume so it decreases inversely proportional to the square of the mass. The density of supermassive black holes can even be less than that of water.

Do black holes evaporate?

Yes, but very slowly. Hawking radiation consumes energy and mass over time, but for black holes the size of stars their lifetime is much longer than the age of the universe. Only tiny black holes evaporate quickly, and black holes with temperatures lower than that of the cosmic microwave background are not currently shrinking, they're growing.

Black Hole Calculator

Black hole

Mass

Schwarzschild radius (event horizon)

Enter either a mass or a Schwarzschild radius to reveal the black hole's properties.

Estimate the tidal force (spaghettification)

See how hard the black hole would stretch an object at its event horizon.

Please input values into the fields

What this tells you

Every value assumes a non-rotating, uncharged (Schwarzschild) black hole, fully described by its mass alone. Real black holes spin, which shrinks the horizon a little.

The black hole calculator tool converts a single numerical value, the mass, into a comprehensive physical picture of a black hole. By entering the mass, the size of the event horizon, the density of the celestial body, the strength of gravity, the temperature of radiation and the time until evaporation are displayed. Reverse calculations are also possible: If the Schwarzschild radius is entered, then the corresponding mass results.

What the Schwarzschild Radius Means:

The Schwarzschild radius is the event horizon, i.e., the limit beyond which nothing can escape. At this distance, the escape velocity reaches the speed of light, so that neither matter nor light can get out.

Karl Schwarzschild discovered this radius in 1916, just a few months after Albert Einstein published his general theory of relativity. For a non-spinning and uncharged black hole, the size is determined only by its mass, and it doesn't depend on anything else. Any object that gets compressed to a volume equal to its own Schwarzschild radius becomes a black hole.

How to use this calculator tool:

Select the known information. In mass mode, enter the mass and select a unit. You can choose between kilograms, Earth masses, or solar masses. The calculator tool will calculate the event horizon and all derived properties.

In the radius mode, if you enter in the schwarzschild radius, it will calculate the mass that would create that radius. The tidal forces option allows you to check what force a black hole exerts on an object of a certain size falling into it.

Calculation formula:

The radius of the event horizon is derived directly from general relativity.

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

where G is the gravitational constant, M is mass and c is the speed of light. As the horizon encloses a sphere, its surface area and average density within are calculated as follows:

A = 4\pi r_s^2 = \frac{16\pi G^2 M^2}{c^4} \qquad \rho = \frac{3\,c^6}{32\pi G^3 M^2}

The average density decreases with increasing mass. The density of a stellar black hole is higher than that of atomic nuclei, while the density of a supermassive black hole may be less than that of air. The Newtonian surface gravity at the event horizon is given by:

g = \frac{c^4}{4\,G\,M}

In 1974 Stephen Hawking proved that black holes are not completely black objects. Because of quantum mechanical effects at the event horizon, black holes have a temperature, a faint glow and a finite lifetime.

T = \frac{\hbar\,c^3}{8\pi k_B\,G\,M} \qquad L = \frac{\hbar\,c^6}{15360\,\pi G^2 M^2}

t_{\text{evap}} = \frac{5120\,\pi\,G^2 M^3}{\hbar\,c^4}

The hotter and smaller a black hole is, the faster it evaporates. This is because temperature increases as mass decreases. As Bekenstein and Hawking showed, entropy is proportional to the area of the event horizon rather than the volume. This is one of the most important clues for understanding quantum gravity.

\frac{S}{k_B} = \frac{4\pi\,G\,M^2}{\hbar\,c}

Example calculation:

If we take the Sun as an example, its mass is about 1.989 multiplied by 10 to the power of 30 kilograms. If we put this value in the formula for the radius, we get approximately 2,954 meters. So the diameter of the event horizon of the Sun if it actually became a black hole would be less than 3 km.

If we extend the scale to ten times the mass of the Sun, we get a typical stellar-mass black hole with a radius of about 29.5 km proportional to that. Since the radius is directly proportional to the mass, doubling the mass doubles the event horizon.

Black holes of various mass scales:

With the same calculation we can cover a wide range of mass scales from hypothetical primordial black holes to the huge monsters in the center of galaxies.

Object	Approximate mass	Schwarzschild radius
Primordial black hole	1 trillion kg	about 1.5 femtometres
Earth	5.97 times ten to the 24th kg	about 8.9 millimetres
The Sun	1.989 times ten to the 30th kg	about 2.95 km
Cygnus X-1	about 21 solar masses	about 62 km
Sagittarius A*	about 4.3 million solar masses	about 12.7 million km
M87*	about 6.5 billion solar masses	about 128 au

Spaghetti-ization and tidal forces.

The pull of a black hole on an object is stronger on the side that's closer than it is on the far side. This difference in force causes anything falling into the black hole to stretch out. Physicists have given this effect the humorous name "spaghettification."

The degree of this stretching increases with increasing mass but is dramatically reduced by the inverse relationship to the cube of the radius. Tidal forces are extremely strong at the event horizon near small stellar-mass black holes. However, tidal forces are relatively weak near supermassive black holes because the distance from the event horizon is so large. Theoretically, one could cross the event horizon without noticing anything.

Will black holes last forever?

Not quite. As they lose energy through Hawking radiation, isolated black holes slowly lose mass and eventually evaporate. The problem is the timescale. It would take about 10 to the power of 67 years for a black hole with the mass of our sun to disappear. This is a timescale that far exceeds the current age of the universe and is barely imaginable.

There is another threshold. Black holes with temperatures below the value of the cosmic microwave background radiation of 2.725 K absorb more radiation than they emit. Therefore, large black holes are currently growing and not shrinking.

This calculator is for learning and exploration purposes only. It models an idealized non-rotating black hole of the Schwarzschild type, so results may differ from actual rotating black holes.

Frequently asked questions

How is the Schwarzschild Radius Calculated?: You multiply the gravitational constant G by two, then you multiply it by the mass and divide that result by the square of the speed of light. The event horizon diameter for a black hole with one solar mass is about 2.95 km, where the radius is proportional to the mass.
What is Earth's Schwarzschild Radius?: It is about 8.9 millimeters in size. A black hole only forms when the whole Earth is compressed to the size of a marble.
Is a black hole itself smaller than its Schwarzschild radius?: Yes. The schwarzschild radius is the event horizon or border surrounding a black hole. It's assumed that the mass itself collapses into a much smaller region.
Why do larger black holes have lower density?: As the radius increases with mass, the volume increases in proportion to the cube of the mass while the increase in mass is linear. Density is defined as mass divided by volume so it decreases inversely proportional to the square of the mass. The density of supermassive black holes can even be less than that of water.
Do black holes evaporate?: Yes, but very slowly. Hawking radiation consumes energy and mass over time, but for black holes the size of stars their lifetime is much longer than the age of the universe. Only tiny black holes evaporate quickly, and black holes with temperatures lower than that of the cosmic microwave background are not currently shrinking, they're growing.

Related calculators

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Orbital Period CalculatorCalculate orbital period with Kepler's Third Law. Enter any two of the orbital radius, central mass, and period to solve for the third, with orbital and escape velocity, elliptical orbits, and altitude.

Antipode CalculatorFind the antipode of any point on Earth. Enter a latitude and longitude to get the exact opposite coordinates, the distance through and around the planet, and the DMS format.

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 and derivation of the event-horizon radius.
Wikipedia: Hawking radiation
Hawking temperature, luminosity, and evaporation time for a Schwarzschild black hole.
NASA: Black Holes
Overview of stellar, intermediate, and supermassive black holes.
Wikipedia: Bekenstein-Hawking entropy (Black hole thermodynamics)
Entropy proportional to horizon area and the laws of black hole mechanics.