To a first-order approximation, the length is 1/4 wavelength per leg. However, that assumes an ideal conductor in free space. We rarely have the luxury to suspend an antenna a few wavelengths from the ground, water, buildings, and anything else remotely conductive. The way the antenna interacts with those things in the near-field affects its impedance and makes it behave electrically longer.
The common formula you'll see is: Total Length (in feet) = 468 / f (in MHz). If we suppose exactly half wavelength and do the unit conversions, we would expect L (ft) = 492 / f (MHz). So why do the calculators use the shorter length? It's an empirical compromise. Taking 5-10% off total length is generally what's needed to account for things in the near-field. The 468 number has been repeated enough that it's stuck. In practice, I almost always cut dipoles for a full half-wavelength, hook them up to an antenna analyzer and then trim them down. With as cheap as hookup wire is, I'd rather not take the risk of being too short and having to field solder a splice (not fun on ARRL Field Day).
There's not really a general formula for finding length. The physics is nothing more than Maxwell's equations, but many of the deviations from an ideal dipole come from interactions with the environment. It's difficult to measure and/or predict how the environment will behave, so you're often better off building the antenna and then adjusting it in place. And so we end up with empirical rules of thumb like L = 468 / f.
Antennas are one of the harder topics for amateurs, for sure. The theory is well-understood, sitting somewhere at the intersection of EE and physics. I'm lucky enough to have a strong background in both fields, but there's a clear lack of curriculum for amateurs without that background. This website at least seems useful for building intuition, so hopefully it helps you some.
The common formula you'll see is: Total Length (in feet) = 468 / f (in MHz). If we suppose exactly half wavelength and do the unit conversions, we would expect L (ft) = 492 / f (MHz). So why do the calculators use the shorter length? It's an empirical compromise. Taking 5-10% off total length is generally what's needed to account for things in the near-field. The 468 number has been repeated enough that it's stuck. In practice, I almost always cut dipoles for a full half-wavelength, hook them up to an antenna analyzer and then trim them down. With as cheap as hookup wire is, I'd rather not take the risk of being too short and having to field solder a splice (not fun on ARRL Field Day).
There's not really a general formula for finding length. The physics is nothing more than Maxwell's equations, but many of the deviations from an ideal dipole come from interactions with the environment. It's difficult to measure and/or predict how the environment will behave, so you're often better off building the antenna and then adjusting it in place. And so we end up with empirical rules of thumb like L = 468 / f.
Antennas are one of the harder topics for amateurs, for sure. The theory is well-understood, sitting somewhere at the intersection of EE and physics. I'm lucky enough to have a strong background in both fields, but there's a clear lack of curriculum for amateurs without that background. This website at least seems useful for building intuition, so hopefully it helps you some.