We can use dimensional analysis to construct a length and a mass scale using Planck's constant, , Newton's universal gravitational constant, G, and the speed of light, c.
Planck's constant has units of: length2 mass time
Newton's constant has units of: length3 (mass time2)
The speed of light has units of: length/time
We can construct a new constant that has units of length as follows:
The above expression has units of:
If we instead combine , G, and c into a new constant with units of mass, we find:
When we plug in actual numbers, we find that the Planck length is 1.61 x 10-35 m, 1020 times smaller than the nucleus of an atom. The Planck mass (which, of course, also determines an energy scale) is around 10-5 g, which is equivalent to ~ 1019 GeV. These length and mass scales help physicists estimate when it's necessary to do a calculation using a quantum theory of gravity.