Schwarzschild Radius Calculator
Compute the Schwarzschild radius — the event horizon size — of a black hole given its mass in solar masses. Use it to visualize how compact an object must be to become a black hole.
About this calculator
The Schwarzschild radius is the critical radius at which an object's escape velocity equals the speed of light, making it a black hole. It is derived from general relativity: r_s = 2GM / c², where G = 6.674 × 10⁻¹¹ N m² kg⁻² is the gravitational constant, M is the mass in kilograms, and c = 2.998 × 10⁸ m/s is the speed of light. This calculator accepts mass in solar masses (M☉ = 1.989 × 10³⁰ kg), converts internally, and returns the radius in kilometers. The result scales linearly with mass: a black hole ten times more massive has an event horizon ten times larger. Our Sun would need to be compressed to a sphere of radius ~3 km to become a black hole, while a stellar-mass black hole of 10 M☉ has r_s ≈ 29.5 km.
How to use
Calculate the Schwarzschild radius of a 10 solar-mass black hole. Step 1 – Convert mass: M = 10 × 1.989 × 10³⁰ = 1.989 × 10³¹ kg. Step 2 – Apply formula: r_s = (2 × 6.674 × 10⁻¹¹ × 1.989 × 10³¹) / (2.998 × 10⁸)² = (2.655 × 10²¹) / (8.988 × 10¹⁶) ≈ 29,540 m ≈ 29.5 km. Step 3 – Divide by 1,000 for km: ≈ 29.5 km. Enter 10 M☉ and the calculator returns ~29.5 km instantly. For the supermassive black hole in M87 (6.5 × 10⁹ M☉) this scales to about 19.2 billion km.
Frequently asked questions
What is the Schwarzschild radius and what does it represent physically?
The Schwarzschild radius (r_s) is the radius of the event horizon of a non-rotating black hole — the boundary beyond which nothing, not even light, can escape. It is not a physical surface but a mathematical boundary in spacetime. Any object compressed below its own Schwarzschild radius will inevitably collapse into a singularity. For everyday objects the values are tiny: Earth's Schwarzschild radius is about 8.9 mm, and a person's would be smaller than a proton.
How does the Schwarzschild radius change with black hole mass?
The Schwarzschild radius scales linearly with mass: r_s = 2GM/c². Double the mass and the event horizon doubles in size. This contrasts with ordinary stars, whose radii grow much more slowly with mass. A solar-mass black hole has r_s ≈ 3 km; the black hole at the center of the Milky Way (Sgr A*, ~4 million M☉) has r_s ≈ 12 million km, roughly 17 times the Sun's actual radius.
Why is the speed of light squared in the Schwarzschild radius formula?
The c² term comes directly from Einstein's field equations of general relativity, which relate spacetime curvature to mass-energy. The escape velocity from the surface of a sphere of mass M and radius r is v_esc = √(2GM/r); setting v_esc = c and solving for r gives r_s = 2GM/c². The c² in the denominator means that because light travels so fast, an enormous amount of mass packed into a very small volume is needed to trap it, which is why stellar-mass black holes are only tens of kilometers across despite containing more mass than the Sun.