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: length^{2} mass time

Newton's constant has units of: length^{3} (mass time^{2})

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, 10^{20} 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 ~ 10^{19} GeV. These length and mass scales help physicists estimate when it's necessary to do a calculation using a quantum theory of gravity.