Schwarzschild Radius Calculator

What the Schwarzschild radius calculator does

The Schwarzschild radius is the size a mass would have to be squeezed into to become a black hole. This calculator finds it from the mass, giving the radius of the event horizon, and works for anything from planets to stars to supermassive black holes.

Below is what the Schwarzschild radius is, the equation behind it, what the event horizon means, and a worked example.

How to use it

1. Enter the mass, in kilograms, solar masses, Earth masses, or other units.
2. Choose the unit for the result, from metres to astronomical units.
3. Press Calculate for the Schwarzschild radius, or Reset to clear it.

What the Schwarzschild radius is

The Schwarzschild radius is a special distance associated with any mass: it is the radius at which, if all that mass were compressed inside it, the object would become a black hole. Named after Karl Schwarzschild, who derived it from Einstein's general relativity in 1916, it marks the boundary where gravity becomes so overwhelming that not even light can escape. For an ordinary object, the Schwarzschild radius is fantastically small compared with its real size, so nothing happens. But squeeze the mass within that radius, and a black hole forms.

The idea connects directly to escape velocity, the speed needed to break free of a body's gravity. The Schwarzschild radius is precisely the distance at which the escape velocity equals the speed of light. Since nothing can travel faster than light, nothing can escape from within it, which is what makes a black hole black. This calculator computes that critical radius for any mass, revealing how small a given object would have to become to cross over into a black hole.

The equation it uses

The Schwarzschild radius depends only on the mass, through the gravitational constant and the speed of light:

r_s = 2 G M ÷ c²

Here r_s is the Schwarzschild radius, G is the gravitational constant, M is the mass, and c is the speed of light. The radius is directly proportional to the mass, so doubling the mass doubles the Schwarzschild radius. Because the speed of light squared is an enormous number in the denominator, the radius comes out tiny for everyday masses, which is why black holes require such extraordinary compression to form. The calculator evaluates this formula and converts the result to your chosen unit.

The event horizon and the point of no return

The sphere defined by the Schwarzschild radius is called the event horizon, and it is one of the most dramatic concepts in physics. It is a one-way boundary: anything that crosses it, matter, light, information, can never come back out. From outside, you can never see what happens within, because no signal can climb back across the horizon to reach you. It is, quite literally, a point of no return, the edge beyond which the rest of the universe loses all contact.

Importantly, the event horizon is not a solid surface but a boundary in space. An unlucky traveller falling in would not hit anything as they crossed it; they would simply pass a line beyond which escape is impossible, drawn inexorably toward the centre. The event horizon also sets the apparent size of a black hole, since it is the closest we can ever observe. The famous images of black holes show light bending around just outside this horizon, whose radius is exactly what this calculator computes.

From a marble-sized Earth to giant black holes

The Schwarzschild radius gives a vivid sense of how extreme black holes are. For the Earth, it is just 9 millimetres: our entire planet would have to be crushed to the size of a marble to become a black hole. For the Sun, it is about 3 kilometres, meaning the Sun, more than a million kilometres across, would need to be compressed to the size of a small town. Such compression does not happen to ordinary stars, which is why they do not collapse into black holes.

At the other end of the scale lie supermassive black holes, lurking at the centres of galaxies, with masses of millions or billions of Suns. Their Schwarzschild radii grow in proportion, reaching across distances comparable to planetary orbits or beyond. The black hole at the centre of our own galaxy, some four million solar masses, has a Schwarzschild radius of roughly a tenth of an astronomical unit. Curiously, because the radius grows with mass while volume grows with its cube, the largest black holes are actually less dense than water. The calculator spans this whole range, from the marble-sized Earth to monsters larger than the Solar System.

Units and precision

The calculator takes the mass in kilograms, grams, pounds, Earth masses, or solar masses, and returns the Schwarzschild radius in your choice of length units, from metres up to light-years, which suits the enormous span from small bodies to supermassive black holes. The calculation is an exact application of the formula. The radius is the meaningful output and is reported in whichever length unit you select.

A worked example

Take the Sun, with a mass of about 1.989 × 10³⁰ kilograms.

The Schwarzschild radius is r_s = 2 G M ÷ c² ≈ 2.95 kilometres. So if the entire Sun were somehow compressed into a sphere just under 3 kilometres across, it would become a black hole, with that radius as its event horizon. The Earth, far less massive, has a Schwarzschild radius of only about 9 millimetres, the size of a marble, showing how staggeringly dense an object must be to cross this threshold.

Questions people ask

How do you calculate the Schwarzschild radius?

Use r_s = 2GM/c², where G is the gravitational constant, M is the mass, and c is the speed of light. The radius is proportional to the mass.

What is the event horizon?

The sphere at the Schwarzschild radius, a one-way boundary. Anything crossing it, including light, can never escape, which is why it is called the point of no return.

Why can't light escape a black hole?

Because at the Schwarzschild radius the escape velocity equals the speed of light. Since nothing can travel faster than light, nothing can escape from within the event horizon.

What is Earth's Schwarzschild radius?

About 9 millimetres. Our whole planet would have to be compressed to the size of a marble to become a black hole, which shows how extreme the required density is.

References

A quick note on where the physics comes from. The Schwarzschild radius and the event horizon are standard general relativity, set out in OpenStax's University Physics and in Georgia State University's HyperPhysics. NASA describes black holes and their horizons. The HyperPhysics link is worth a quick click to confirm it lands where you expect.

1. OpenStax, University Physics Volume 1, Section 13.7, Einstein's Theory of Gravity. https://openstax.org/books/university-physics-volume-1/pages/13-7-einsteins-theory-of-gravity
2. HyperPhysics, Black Holes and the Schwarzschild Radius. http://hyperphysics.phy-astr.gsu.edu/hbase/Astro/blkhol.html
3. NASA, Black Holes. https://science.nasa.gov/universe/black-holes/