Why is an atom the size it is? Balance the electron’s quantum ‘spread’ against its electrical pull to the nucleus and a length falls out with no measurement: the BOHR RADIUS, 5.3×10⁻¹¹ metres. It sets the scale of all of chemistry — the size of matter you can touch. Slide to build it from the constants.
The Bohr radius a₀ = 4πε₀ħ²/(mᵉe²) ≈ 5.29×10⁻¹¹ m is the natural length of the hydrogen atom — where quantum kinetic energy (which grows as you confine the electron) balances the Coulomb attraction (which pulls it in). It falls out of ħ, the electron mass mᵉ, its charge e, and ε₀; equivalently a₀ = ħ/(mᵉc·α). It is the yardstick of chemistry and condensed matter. A fail-loud self-check throws unless the assembled value lands between 5.0 and 5.6 ×10⁻¹¹ m. ◆ real physics, node-verified.
Exact from the constants for hydrogen; real atoms and orbitals differ, but a₀ sets the SCALE. Another unit that falls out of a balance, no ruler applied.