⚫ Schwarzschild Radius Calculator
Calculate the Schwarzschild radius (event horizon) of a black hole from its mass using the formula Rs = 2GM/c². Solve for radius, mass, or density. Includes real-world comparisons across atomic, human, planetary, stellar, and galactic scales.
What Is the Schwarzschild Radius?
The Schwarzschild radius (Rs) is the radius of the event horizon of a non-rotating black hole. It defines the boundary beyond which nothing — not even light — can escape the black hole's gravitational pull. Any object compressed within its Schwarzschild radius becomes a black hole.
It is named after the German physicist Karl Schwarzschild, who derived the first exact solution to Einstein's field equations of general relativity in 1916, just months after Einstein published his theory. The solution describes the spacetime geometry around a spherical, non-rotating mass.
The Schwarzschild Radius Formula
The formula for the Schwarzschild radius is:
Rs = 2GM / c²
where G = 6.67430 × 10⁻¹¹ m³ kg⁻¹ s⁻² (gravitational constant), M = mass of the object (kg), c = 299,792,458 m/s (speed of light)
A useful simplification for solar masses is:
Rs ≈ 2.95 km × (M / M☉)
This means a black hole of 1 solar mass has a Schwarzschild radius of about 2.95 km.
Key Properties
- Linear with mass: Rs is directly proportional to mass — double the mass, double the radius.
- Inversely squared density: The average density inside the event horizon decreases as 1/M². Stellar-mass black holes are incredibly dense, while supermassive black holes have surprisingly low average densities.
- Independent of composition: Any mass compressed within its Schwarzschild radius becomes a black hole, regardless of what it's made of.
- Event horizon: The Schwarzschild radius is not a physical surface — it is a mathematical boundary in spacetime.
Known Schwarzschild Radii
Here are some notable Schwarzschild radii:
- Sun (1 M☉): ≈ 2.95 km
- Earth (1 M⊕): ≈ 8.87 mm (about the size of a marble)
- Jupiter (1 M♃): ≈ 2.82 m
- Human (70 kg): ≈ 1.04 × 10⁻²³ cm (far smaller than an atom)
- Sagittarius A* (4.3 million M☉): ≈ 1.27 × 10⁷ km (about 0.085 AU)
- M87* (6.5 billion M☉): ≈ 1.92 × 10¹⁰ km (about 128 AU)
Why Use This Calculator?
- Solve for any variable: Calculate Rs, mass, or average density depending on what's unknown.
- Multiple units: Enter mass in solar masses, kilograms, Earth masses, or Jupiter masses; output radius in km, m, cm, mm, miles, AU, or light-years.
- Scale classification: Automatically classifies the result as atomic, human, planetary, stellar, or galactic scale.
- Step-by-step solution: See exactly how the calculation is performed.
- Free & private: No registration, no data storage — all calculations are done in your browser.
❓ Schwarzschild Radius Calculator FAQ
What is the Schwarzschild radius?
The Schwarzschild radius is the radius of the event horizon of a non-rotating black hole. It is the distance from the singularity at which the escape velocity equals the speed of light.
What is the formula for the Schwarzschild radius?
The formula is Rs = 2GM/c², where G is the gravitational constant, M is mass, and c is the speed of light. For solar masses, Rs ≈ 2.95 km × (M/M☉).
What is the Schwarzschild radius of the Sun?
The Sun's Schwarzschild radius is approximately 2.95 km. If the Sun were compressed to this radius, it would become a black hole.
What is the Schwarzschild radius of Earth?
Earth's Schwarzschild radius is approximately 8.87 mm — about the size of a marble. If Earth were compressed to this size, it would become a black hole.
Can any object become a black hole?
In principle, yes — any mass compressed within its Schwarzschild radius would become a black hole. However, in practice, only massive stars (above about 3 solar masses) can collapse into black holes at the end of their lives.
What is the difference between Schwarzschild radius and event horizon?
The Schwarzschild radius is the numerical radius. The event horizon is the actual boundary in spacetime at that radius. They refer to the same thing but from different perspectives.
What is the density of a black hole?
The average density inside the event horizon is ρ = 3c⁶ / (32πG³M²). It decreases as 1/M². Stellar-mass black holes are incredibly dense, while supermassive black holes have surprisingly low average densities — some are less dense than water.
Does the Schwarzschild radius apply to rotating black holes?
No. The Schwarzschild solution assumes a non-rotating, uncharged black hole. For rotating black holes, the Kerr metric must be used, which has two horizons (outer and inner) and a different radius.
How accurate is this calculator?
This calculator uses the exact Schwarzschild formula with CODATA-recommended values for G, c, and the solar mass. Results are accurate to the precision you select.
Is this calculator free?
Yes, this calculator is completely free to use. No registration or personal data storage is required.