Schwarzschild Radius Calculator
The event horizon radius for a given mass, in metres and kilometres, with the mean density inside it.
Results
What this tool does
Compress anything far enough and the escape velocity at its surface reaches the speed of light; the radius where that happens is named after Karl Schwarzschild, who found the solution in 1916 while serving on the Russian front. The relationship is beautifully simple — the radius is directly proportional to the mass — which means a black hole of a billion solar masses is a billion times wider than one of a single solar mass, and vastly less dense.
Formula
rs = 2 G M / c²
Variables
| Symbol | Meaning | Unit |
|---|---|---|
mm | Mass | kg |
RS | Schwarzschild radius | m |
RK | Schwarzschild radius | km |
SM | Mass in solar masses | — |
DE | Mean density inside | kg/m³ |
Worked example
- Mass1.989e+30 kg
- Schwarzschild radius2954.126555 m
- Schwarzschild radius2.954127 km
- Mass in solar masses1.000000
- Mean density inside1.84187e+19 kg/m³
Limitations
- The formula assumes ideal conditions: no friction losses, no air resistance and no efficiency losses unless you enter them.
Frequently asked questions
Does every object have a Schwarzschild radius?
Mathematically yes, but it only means something if the object is actually smaller than it. The Earth's is about 9 millimetres and the Sun's is about 3 kilometres, and since neither is anywhere near that compact, neither has an event horizon. The radius is simply the size at which the escape velocity would reach the speed of light.