The fundamental length scale derived from Planck’s constant, gravity, and light speed is the Planck length.
The Planck length is defined by combining three fundamental constants: the reduced Planck constant, the gravitational constant, and the speed of light. Its formula is ℓₚ = √(ℏG/c³), giving a value of approximately 1.616 × 10⁻³⁵ meters. That is vastly smaller than an atomic nucleus, whose scale is around 10⁻¹⁵ meters.
German physicist Max Planck introduced natural units in 1899, seeking a system based on universal constants rather than human-made measurements. The Planck length is one member of this system, alongside Planck time, Planck mass, and Planck temperature.
It is often described as the scale at which quantum mechanics and general relativity are both expected to matter significantly. However, physics has not established that space is definitely made of discrete pixels of Planck-length size. The Planck length is a theoretically important scale, not a directly observed smallest possible distance.
The Bohr radius describes the typical size of a hydrogen atom, while the Compton wavelength relates a particle’s mass to quantum wavelength. Neither combines gravity with quantum mechanics in the same way.