Schwarzschild Radius Calculator

Find the event horizon radius of a black hole for any mass, or find the mass needed to form a black hole of a given radius.

⚫ Schwarzschild Radius Calculator
Object mass1.000 M☉
M☉
Schwarzschild radius3.000 km
km
Schwarzschild Radius
—
In Metres
—
Mean Collapse Density
—
Implied Mass
—
In Kilograms
—

⚫ What is the Schwarzschild Radius?

The Schwarzschild radius is the critical radius to which a mass must be compressed for the escape velocity at its surface to equal the speed of light. Once any object is compressed within this radius, it collapses into a black hole and its boundary becomes the event horizon. The formula was first derived by Karl Schwarzschild in December 1915, just weeks after Einstein published general relativity.

The Schwarzschild radius appears in many practical calculations. Astrophysicists use it to characterise black hole sizes, predict light-bending near compact objects, calculate gravitational redshift, and determine the innermost stable circular orbit (which sits at 3 times the Schwarzschild radius for a non-rotating black hole). The Event Horizon Telescope imaging of M87* and Sgr A* resolved the emission region on scales of a few Schwarzschild radii.

A common point of confusion is density. Because the Schwarzschild radius grows linearly with mass while volume grows as the cube of radius, the mean density inside the Schwarzschild sphere falls rapidly with mass. A stellar-mass black hole of 10 solar masses has mean density 100 times nuclear density, but a billion solar mass black hole has a mean density lower than water. You do not need exotic super-dense material to form the most massive black holes in the universe.

The Schwarzschild radius also applies to everyday objects, though at scales that are physically irrelevant. Earth's Schwarzschild radius is about 8.87 mm. A 70 kg person has a Schwarzschild radius of roughly 10^-25 m, far smaller than a proton. These numbers illustrate why everyday gravity is far too weak to create black holes from ordinary matter, and why only the collapse of massive stellar cores or ultra-high-energy particle collisions could potentially do so.

📐 Formula

rs  =  2GM ÷ c2  ≈  2.954 × (M / M☉) km
G = gravitational constant = 6.674 × 10−11 m3 kg−1 s−2
M = mass of the object (kg)
c = speed of light = 2.998 × 108 m/s, so c2 = 8.988 × 1016 m2/s2
Mean density inside rs: ρ = 3c6 / (32π G3 M2)
Example: For M = 10 M☉, rs = 2 × 6.674e−11 × 10 × 1.989e30 / 8.988e16 = 29,538 m ≈ 29.5 km

📖 How to Use This Calculator

Steps

1
Select a mode — choose Find Radius to compute the Schwarzschild radius from a mass, or Find Mass to find the mass implied by a known event horizon radius.
2
Enter the mass or radius — in Find Radius mode, type a mass in solar masses. In Find Mass mode, enter a radius in kilometres.
3
Read the results — the calculator shows the radius in human-readable units (km for stellar black holes, AU for supermassive ones), the raw value in metres, and the mean density inside the Schwarzschild sphere.

💡 Example Calculations

Example 1 — Solar Mass Black Hole (1 M☉)

A black hole with the mass of the Sun

1
rs = 2 × 6.674 × 10−11 × 1.989 × 1030 / (2.998 × 108)2 = 2,953.8 m.
2
Mean density at rs: ρ = 1.989 × 1030 / (4/3 × π × 2953.83) = 1.842 × 1019 kg/m3 (about 9 times nuclear density).
rs = 2.954 km (2.954 × 103 m), mean density = 1.842 × 1019 kg/m3
Try this example →

Example 2 — 10 Solar Mass Stellar Black Hole

A typical stellar-mass black hole: 10 M☉

1
rs = 2.954 km × 10 = 29.538 km (radius scales linearly with mass).
2
Density scales as 1/M2: ρ = 1.842 × 1019 / 100 = 1.842 × 1017 kg/m3.
rs = 29.538 km (2.954 × 104 m), mean density = 1.842 × 1017 kg/m3
Try this example →

Example 3 — Sgr A* (4 × 106 M☉)

The Milky Way's supermassive black hole

1
rs = 2.954 km × 4 × 106 = 11,815,381 km.
2
Convert to AU: 11,815,381 km / 149,600,000 km/AU = 0.079 AU (smaller than Mercury's perihelion distance).
rs = 0.079 AU (1.182 × 1010 m), as imaged by the Event Horizon Telescope
Try this example →

Example 4 — Find Mass from a 3 km Radius

Reverse: what mass produces a 3 km Schwarzschild radius?

1
Switch to Find Mass mode. Enter radius = 3 km.
2
M = rs × c2 / (2G) = 3000 × 8.988 × 1016 / (2 × 6.674 × 10−11) = 2.020 × 1030 kg.
Mass = 1.0156 M☉ (2.020 × 1030 kg)
Try this example →

❓ Frequently Asked Questions

What is the Schwarzschild radius formula?+
The Schwarzschild radius is r_s = 2GM/c^2. In convenient units, r_s in kilometres equals approximately 2.954 times the mass in solar masses. For SI units: r_s (m) = 1.485 × 10^-27 × M (kg).
What is the Schwarzschild radius of Earth?+
Earth's Schwarzschild radius is about 8.87 mm. Compressing the entire mass of Earth (5.972 × 10^24 kg) into a sphere with a diameter of less than 2 centimetres would create a black hole. In practice this requires pressures far beyond anything naturally occurring.
How does the Schwarzschild radius relate to the event horizon?+
For a non-rotating (Schwarzschild) black hole, the event horizon is exactly at the Schwarzschild radius. For a rotating (Kerr) black hole, the event horizon radius is smaller, lying between r_s/2 and r_s depending on spin. This calculator computes the Schwarzschild (non-rotating) case.
Why do supermassive black holes have such low densities?+
Because density scales as M/r_s^3 and r_s scales as M, the density scales as 1/M^2. A billion solar mass black hole has a Schwarzschild radius of about 3 billion km and a mean density of roughly 20 kg/m^3 inside that sphere, less than water. Falling through the event horizon of such a black hole would be a gentle experience.
Can I calculate the Schwarzschild radius for a proton or neutron?+
Yes. A proton has mass 1.673 × 10^-27 kg, giving r_s = 2 × 6.674e-11 × 1.673e-27 / (9e16) = 2.48 × 10^-54 m. This is 10^19 times smaller than the Planck length, far outside the domain of classical general relativity. Quantum gravity effects would dominate at such scales.
What is the innermost stable circular orbit (ISCO) for a Schwarzschild black hole?+
The ISCO for a non-rotating Schwarzschild black hole is at 3r_s = 6GM/c^2. Inside this orbit, no stable circular path exists and matter spirals inward. For a 10 solar mass black hole, the ISCO is at about 88.6 km. For maximally spinning Kerr black holes, the ISCO can reach as close as 0.5r_s.
How was the Schwarzschild radius first calculated?+
Karl Schwarzschild derived it in December 1915 by finding the exact vacuum solution to Einstein's field equations for a spherically symmetric mass. He sent the solution to Einstein from the Eastern Front during World War I. Einstein presented it to the Prussian Academy in January 1916, weeks before Schwarzschild died of an illness contracted at the front.
Does the Schwarzschild radius formula apply inside matter (like a neutron star)?+
The Schwarzschild metric describes the vacuum spacetime outside a spherical mass. Inside matter, the metric is different (the Tolman-Oppenheimer-Volkoff solution governs neutron stars). However, if the stellar radius drops below r_s, collapse to a black hole is inevitable regardless of the equation of state of the matter.
What is the Schwarzschild radius of Betelgeuse?+
Betelgeuse has a mass of about 12 to 20 solar masses. Using 16 solar masses, r_s = 2.954 × 16 = 47.3 km. If Betelgeuse collapses into a black hole at the end of its life, the remnant's event horizon would be about 47 km across, roughly the size of a small city.
How does gravitational lensing relate to the Schwarzschild radius?+
The photon sphere, where photons orbit in unstable circular orbits, is at 1.5 r_s = 3GM/c^2. The Einstein ring radius seen by a distant observer and the shadow size observed by the Event Horizon Telescope are both proportional to r_s. A larger r_s means a larger lensing cross-section and a larger observable shadow.
Tags: Schwarzschild Radius Black Hole Radius Event Horizon Calculator Formula Free Online General Relativity

🔗 Related Calculators

💡
Eddington Luminosity Calculator
Science - Astrophysics
🌡️
Hawking Radiation Temperature Calculator
Science - Astrophysics
💥
Chandrasekhar Limit Calculator
Science - Astrophysics
🌀
TOV Limit Estimator
Science - Astrophysics

What is the Schwarzschild radius of the Sun?

The Schwarzschild radius of the Sun is approximately 2.95 km. The Sun would have to be compressed into a sphere less than 3 km across to become a black hole. In reality the Sun will never reach this density, ending its life as a white dwarf.

How is the Schwarzschild radius calculated?

The formula is r_s = 2GM/c^2, where G = 6.674e-11 m^3 kg^-1 s^-2 is Newton's constant, M is the mass in kilograms, and c = 2.998e8 m/s is the speed of light. For M in solar masses, r_s in km equals approximately 2.954 times the mass in solar masses.

What is the Schwarzschild radius of a 10 solar mass black hole?

A 10 solar mass black hole has a Schwarzschild radius of approximately 29.5 km, comparable to the size of a small city. The mean density inside this sphere is 1.84e17 kg per cubic metre, or roughly 100 times nuclear density.

What happens at the Schwarzschild radius?

The Schwarzschild radius marks the event horizon of a non-rotating black hole. Inside this boundary, no signal can escape because the escape velocity equals the speed of light. A distant observer sees infalling matter appear to freeze at the horizon due to gravitational time dilation.

Does every object have a Schwarzschild radius?

Yes. Every mass has a Schwarzschild radius, but only black holes are actually compressed inside theirs. The Schwarzschild radius of a human (mass ~70 kg) is about 1e-25 m, far smaller than an atomic nucleus.

How big is the Schwarzschild radius of Sagittarius A*?

Sgr A* has a mass of about 4 million solar masses, giving a Schwarzschild radius of roughly 11.8 million km or 0.079 AU. The Event Horizon Telescope imaging in 2022 resolved structure on scales comparable to this radius.

Why does a more massive black hole have lower density at its event horizon?

The Schwarzschild radius grows linearly with mass, so the enclosed volume grows as the cube of mass. Mean density (mass divided by volume) therefore falls as 1/M^2. A billion solar mass black hole has a mean density lower than water inside its Schwarzschild sphere.

What is the Schwarzschild radius of Earth?

Earth's Schwarzschild radius is about 8.87 mm, slightly less than a centimetre. Compressing all of Earth into a marble-sized sphere would create a black hole.

📌 Quick Tips

💡The Schwarzschild radius is exactly 2.954 km for a 1 solar mass object. A stellar-mass black hole of 10 M☉ has r_s of about 29.5 km.
💡Earth compressed to a sphere of radius 8.87 mm would become a black hole. The Moon would need to be smaller than 0.11 mm.
💡Supermassive black holes have very low average densities. The Schwarzschild sphere of a billion solar mass black hole is less dense than water.
💡The mean density at the Schwarzschild radius scales as 1/M^2, so more massive black holes have lower densities.
💡For Sgr A* (4 million solar masses), the Schwarzschild radius is about 0.079 AU, smaller than Mercury's orbit.