Schwarzschild Radius Calculator
Calculate the Schwarzschild radius (event horizon) of a black hole from its mass. Enter values for instant results with step-by-step formulas.
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Where r_s = Schwarzschild radius, G = gravitational constant (6.674 x 10^-11 m^3 kg^-1 s^-2), M = mass of the object, c = speed of light (2.998 x 10^8 m/s). This gives the radius at which escape velocity equals the speed of light.
Last reviewed: December 2025
Worked Examples
Example 1: Stellar Black Hole (10 Solar Masses)
Example 2: Sagittarius A* (4 Million Solar Masses)
Background & Theory
The Schwarzschild Radius Calculator applies the following established principles and formulas. Astronomy and space science rely on a set of precisely defined physical relationships that allow distances, sizes, motions, and energies of celestial objects to be calculated from observational data. Kepler's three laws of planetary motion, derived empirically in the early seventeenth century, describe elliptical orbits, equal areas swept in equal times, and the harmonic law Tยฒ = aยณ, where T is the orbital period in Earth years and a is the semi-major axis in astronomical units (AU). This relationship holds for any object orbiting the Sun and can be generalized using Newton's law of gravitation. Distances in astronomy are expressed in multiple units: one light-year equals approximately 9.461 ร 10ยนโต meters, one parsec equals 3.086 ร 10ยนโถ meters or about 3.26 light-years, defined as the distance at which one AU subtends one arcsecond of parallax. Angular size is calculated as ฮธ = 206,265 ร (d / D) arcseconds, where d is the physical diameter and D is the distance. The stellar magnitude system uses Pogson's formula: m1 โ m2 = โ2.5 ร log10(F1 / F2), where F represents flux. Each magnitude step corresponds to a flux ratio of approximately 2.512, meaning a first-magnitude star is 100 times brighter than a sixth-magnitude star. Hubble's Law relates recessional velocity to distance: v = Hโd, where the Hubble constant Hโ is approximately 70 km/s/Mpc. Escape velocity from any body is given by v = โ(2GM/r), yielding 11.2 km/s for Earth. Orbital period for a circular orbit follows T = 2ฯโ(rยณ/GM). Luminosity and distance are linked by the inverse square law: F = L / (4ฯdยฒ). Stars are classified by spectral type using the mnemonic OBAFGKM, corresponding to surface temperatures from approximately 30,000 K (O-type) to under 3,500 K (M-type). Each type reflects characteristic absorption spectra tied to ionization states of elements in the stellar photosphere.
History
The history behind the Schwarzschild Radius Calculator traces back through the following developments. The history of astronomy is one of progressive scale โ each era expanding humanity's conception of the universe's size and structure. The Copernican revolution of 1543, when Nicolaus Copernicus published De revolutionibus orbium coelestium, displaced Earth from the center of the cosmos and placed the Sun at the center of the planetary system. Decades later, Galileo Galilei turned a Dutch-invented telescope toward the sky in 1609, discovering the moons of Jupiter, the phases of Venus, and the cratered surface of the Moon โ observations that provided compelling evidence for the heliocentric model and led to his conflict with the Catholic Church. Johannes Kepler, working from Tycho Brahe's meticulous naked-eye observations, derived his three laws of planetary motion between 1609 and 1619. Isaac Newton unified celestial and terrestrial mechanics with his law of universal gravitation in 1687, explaining the cause behind Kepler's empirical laws and enabling precise prediction of planetary positions. The eighteenth and nineteenth centuries brought systematic sky surveys, stellar parallax measurements, and the discovery that the Milky Way is itself a galaxy among many. Edwin Hubble's 1929 observations using the 100-inch Hooker Telescope at Mount Wilson demonstrated that galaxies are receding from us at velocities proportional to their distance โ the first direct evidence for an expanding universe and the empirical basis for Big Bang cosmology. NASA was founded in 1958 following the Sputnik shock, and the Apollo 11 mission landed humans on the Moon on July 20, 1969. The Hubble Space Telescope, launched in 1990, revolutionized observational astronomy by operating above Earth's atmosphere and producing imagery from ultraviolet to near-infrared wavelengths. The first confirmed exoplanet around a Sun-like star was detected in 1995 by Michel Mayor and Didier Queloz using the radial velocity method. The James Webb Space Telescope, launched in December 2021 and fully operational by 2022, extended infrared observations to probe the earliest galaxies formed after the Big Bang.
Frequently Asked Questions
Formula
r_s = 2GM / c^2
Where r_s = Schwarzschild radius, G = gravitational constant (6.674 x 10^-11 m^3 kg^-1 s^-2), M = mass of the object, c = speed of light (2.998 x 10^8 m/s). This gives the radius at which escape velocity equals the speed of light.
Worked Examples
Example 1: Stellar Black Hole (10 Solar Masses)
Problem: Calculate the Schwarzschild radius, density, and Hawking temperature of a black hole with 10 times the mass of our Sun.
Solution: Mass = 10 x 1.989 x 10^30 kg = 1.989 x 10^31 kg\nr_s = 2 x 6.674e-11 x 1.989e31 / (2.998e8)^2\nr_s = 2.954e20 / 8.988e16 = 29,543 meters = 29.54 km\nDensity = M / (4/3 x pi x r_s^3) = 1.84 x 10^14 kg/m^3\nHawking Temp = hbar x c^3 / (8pi x G x M x k_B) = 6.17 x 10^-9 K
Result: Schwarzschild radius: 29.54 km | Density: 1.84 x 10^14 kg/m^3 | Temp: 6.17 nK
Example 2: Sagittarius A* (4 Million Solar Masses)
Problem: Calculate the event horizon size of the Milky Way's central supermassive black hole at approximately 4 million solar masses.
Solution: Mass = 4 x 10^6 x 1.989e30 = 7.956 x 10^36 kg\nr_s = 2 x 6.674e-11 x 7.956e36 / (2.998e8)^2\nr_s = 1.062e27 / 8.988e16 = 1.181 x 10^10 m = 1.181 x 10^7 km\nIn AU = 1.181e7 / 1.496e8 = 0.0789 AU\nDensity = 1.14 x 10^6 kg/m^3 (about the density of gold)
Result: Schwarzschild radius: 11.81 million km (0.079 AU) | Density: ~gold
Frequently Asked Questions
What is the Schwarzschild radius and what does it represent?
The Schwarzschild radius defines the size of the event horizon of a non-rotating, uncharged black hole. Named after Karl Schwarzschild, who derived this solution to Einstein's field equations in 1916, it represents the critical radius at which the escape velocity equals the speed of light. Any object compressed within its own Schwarzschild radius becomes a black hole from which nothing, not even light, can escape. The formula is remarkably simple: r_s = 2GM/c^2, where G is the gravitational constant, M is the mass, and c is the speed of light. For our Sun, the Schwarzschild radius is approximately 2.95 kilometers, meaning if you compressed the entire mass of the Sun into a sphere less than 3 km in radius, it would become a black hole. For Earth, this critical radius is only about 8.87 millimeters.
How are real black holes different from the Schwarzschild solution?
The Schwarzschild solution describes an idealized, non-rotating, electrically neutral black hole in a vacuum. Real astrophysical black holes almost certainly rotate, making the Kerr metric a more accurate description. Rotating black holes have two important differences: they possess an ergosphere outside the event horizon where spacetime itself is dragged along with the rotation, and their singularity is ring-shaped rather than a point. The event horizon of a Kerr black hole is smaller than the Schwarzschild radius for the same mass, with maximum rotation reducing it by half. Charged black holes are described by the Reissner-Nordstrom metric (non-rotating) or the Kerr-Newman metric (rotating and charged), though astrophysical black holes likely carry negligible net charge. Despite these differences, the Schwarzschild radius remains a useful first approximation and upper bound on the event horizon size.
How accurate are the results from Schwarzschild Radius Calculator?
All calculations use established mathematical formulas and are performed with high-precision arithmetic. Results are accurate to the precision shown. For critical decisions in finance, medicine, or engineering, always verify results with a qualified professional.
Can I use Schwarzschild Radius Calculator on a mobile device?
Yes. All calculators on NovaCalculator are fully responsive and work on smartphones, tablets, and desktops. The layout adapts automatically to your screen size.
Why might my result differ from another tool or reference?
Differences typically arise from rounding conventions, the specific version of a formula (for example, simple vs compound interest), or unit inconsistencies between inputs. Check that both tools are using the same formula variant and the same units. The References section links to the authoritative source behind the formula used here.
How do I get the most accurate result?
Enter values as precisely as possible using the correct units for each field. Check that you have selected the right unit (e.g. kilograms vs pounds, meters vs feet) before calculating. Rounding inputs early can reduce output precision.
References
Reviewed by Daniel Agrici, Founder & Lead Developer ยท Editorial policy