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`M = (R*c^2)/(2*G)`

Enter a value for all fields

The **Mass of a Black Hole **calculator computes the estimated mass based on the radius of the event horizon.

**INSTRUCTIONS**: Choose units and enter the following:

- (
**R**) This is the radius of the event horizon

**Mass of the Black Hole (M):** The calculator returns the mass in Solar Masses (multiple of the mass of our Sun). However, this can be automatically converted to compatible units via the pull-down menu.

The Mass of a Black Hole equation builds on the very simple principle of the gravitational attraction of any two masses to define the radius of a black hole. The calculated radius, **R**, defines the event horizon of a black hole, in that any mass, **M**, that is compressed smaller than this radius, **R**, becomes a black hole.

Given that the escape velocity of a body can be computed as: `v_"(escape)" = sqrt((2*M*G)/R)` we can recognize that this velocity must not exceed the speed of light. Therefore we can re-write the equation as `v_"(escape)" = sqrt((2*M*G)/R) < c`. Rewriting this again to solve for R, which represents the radius of the black hole, we get:

`R < (2 * M * G)/c^2` ; where G is the universal gravitational constant and M is the input mass of the black hole. Interpreting this in terms of a mass compressed by gravity, it tells us any mass, M, with a radius less than R is by definition a black hole.

Therefore, the formula for the mass of a black hole (solving for M) is:

`M = (R*c^2)/(2*G)`

where:

- M = Mass of the black hole
- R = Radius of the Event Horizon
- c = Speed of Light
- G = Universal Gravity Constant.

If we were to calculate the radius of the sun, were the sun's present mass became a black hole, we would get the following radius:

`M_"(SUN)" = 1.989*10^30 kg`

And if we enter this mass into this equation, we find the sun's radius as a black hole would be approximately: 2953.9 meters.

Since the sun's present radius is approximated as 696,342 km, this equation would tell us the sun's radius as a black hole would be about `4.24 *10^-6` of its present size (0.00000624 of its present size, which is about 6 millionths of the sun's present size).

University Physics 12th Edition, Chapter 12, Equation #12.29