The radio horizon formula
For a standard atmosphere the distance to the radio horizon is about 4.12 × √h kilometres, with the antenna height h in metres. A handheld at 1.5 m sees about 5 km; an antenna on a 10 m mast sees about 13 km; one at 100 m sees about 41 km. For a link between two antennas you add the two horizon distances: a mast at 30 m and a handheld at 1.5 m can talk over about 28 km on level ground.
The numbers are larger than the geometric horizon because the atmosphere bends radio waves slightly downwards. Engineers model this with an effective Earth radius of 4/3 of the real one, the famous k = 4/3. In super-refraction k grows and the horizon extends; in sub-refraction it shrinks. Try different values in the radio horizon calculator.
Why raising an antenna works so well
The horizon grows with the square root of the height, so the first few metres are the most valuable: going from 2 m to 10 m more than doubles the range, while going from 50 m to 58 m adds almost nothing. On a hilltop the same rule applies to the height above the surrounding terrain, not above sea level.
Height also reduces the loss caused by obstacles close to the antenna such as buildings and trees. That is why repeaters sit on mountains and towers.
Beyond the horizon
The signal does not stop at the horizon: it bends over the curved Earth (diffraction) and scatters in the troposphere, but it weakens quickly, typically by tens of decibels within a few tens of kilometres. Under certain weather conditions (temperature inversions, ducting) VHF and UHF can travel hundreds of kilometres. See line of sight versus diffraction and test your own setup on the coverage map.
