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The constant comes from his derivation of [Planck’s law](https://en.wikipedia.org/wiki/Planck%27s_law) which describes the spectrum of black body radiation of an object as a function of the temperature.

How the spectrum looked at different temperatures was known from observation for 40 years before before Planck published an explanation of why they look like they do. So the constant was derived from the observational data. It was not just the Plancks constant that was determined from that observation it was the first time that the Boltzmann constant was determined.

The problem was the idea that light only exists in quantified states with the energy of each quantum that we today call photons is E= hv where v is the frequency. From that and other known physics you can derive the light spectrum distribution. This idea is also one of the fundamental part of quantum mechanics development.