The Schwarzschild radius is a parameter in the Schwarzschild solution to Einstein's field equations that corresponds to the radius of a sphere in flat space that has the same surface area as that of the event horizon of a Schwarzschild black hole of a given mass. It is a characteristic quantity that may be associated with any quantity of mass. The Schwarzschild radius was named after the German astronomer Karl Schwarzschild, who calculated this solution for the theory of general relativity in 1916.

The Schwarzschild radius is given as $${\displaystyle r_{\text{s}}={\frac {2GM}{c^{2}}},}$$ where G is the Newtonian constant of gravitation, M is the mass of the object, and c is the speed of light.

In 1916, Karl Schwarzschild obtained an exact solution to the Einstein field equations for the gravitational field outside a non-rotating, spherically symmetric body with mass ${\displaystyle M}$ (see Schwarzschild metric). The solution contained terms of the form $${\displaystyle {\frac {1}{1-{\frac {r_{\mathrm {s} }}{r}}}},}$$ which have singularities at ${\displaystyle r=0}$ and ${\displaystyle r=r_{\text{s}}}$. The parameter ${\displaystyle r_{\text{s}}}$ has come to be known as the Schwarzschild radius. The physical significance of these singularities was debated for decades. It was found that the one at ${\displaystyle r=r_{\text{s}}}$ is a coordinate singularity, meaning that it is an artifact of the particular system of coordinates that was used; while the one at ${\displaystyle r=0}$ is a spacetime singularity and cannot be removed. The Schwarzschild radius is nonetheless a physically relevant quantity, as noted above and below.

The Schwarzschild radius of an object is proportional to its mass. Accordingly, the Sun has a Schwarzschild radius of approximately 3.0 km (1.9 mi), whereas Earth's is approximately 9 mm (0.35 in) and the Moon's is approximately 0.1 mm (0.004 in).

Any object whose radius is smaller than its Schwarzschild radius is called a black hole. The surface at the Schwarzschild radius acts as an event horizon in a non-rotating body (a rotating black hole operates slightly differently). Neither light nor particles can escape through this surface from the region inside, hence the name "black hole".

Black holes can be classified based on their Schwarzschild radius, or equivalently, by their density, where density is defined as mass of a black hole divided by the volume of its Schwarzschild sphere. As the Schwarzschild radius is linearly related to mass, while the enclosed volume corresponds to the third power of the radius, small black holes are therefore much more dense than large ones. The volume enclosed in the event horizon of the most massive black holes has an average density lower than main sequence stars.

A supermassive black hole (SMBH) is the largest type of black hole, though there are few official criteria on how such an object is considered so, on the order of hundreds of thousands to billions of solar masses. (Supermassive black holes up to 21 billion (2.1 × 10¹⁰) M_☉ have been detected, such as NGC 4889.) Unlike stellar mass black holes, supermassive black holes have comparatively low average densities. (Note that a (non-rotating) black hole is a spherical region in space that surrounds the singularity at its center; it is not the singularity itself.) With that in mind, the average density of a supermassive black hole can be less than the density of water.^{[citation needed]}

The Schwarzschild radius of a body is proportional to its mass and therefore to its volume, assuming that the body has a constant mass-density. In contrast, the physical radius of the body is proportional to the cube root of its volume. Therefore, as the body accumulates matter at a given fixed density (in this example, 997 kg/m³, the density of water), its Schwarzschild radius will increase more quickly than its physical radius. When a body of this density has grown to around 136 million solar masses (1.36 × 10⁸ M_☉), its physical radius would be overtaken by its Schwarzschild radius, and thus it would form a supermassive black hole.