How this calculator works
A tangent sightline meets the local radius at a right angle. The observer, centre and tangent point form a right triangle whose sides are R+h, R and the straight sightline distance. Surface distance follows the central angle instead. The model assumes a smooth sphere without atmospheric refraction or terrain. Planet presets use approximate mean radii; giant planets do not have a solid visible surface.
Formula
d = sqrt[h(2R+h)]; h = d²/[sqrt(R²+d²)+R]; dip = atan(d/R); surface arc = R × dip.
Worked example
On a sphere with radius 3 m, an observer 2 m above the surface has a 4 m tangent horizon distance. On Earth, a 2 m eye height gives about 5.048 km.
Advertisement
Frequently asked questions
Which distance is reported?
Both the straight tangent sightline and the shorter surface arc are reported and labelled separately.
Does this include atmospheric bending?
No. Atmospheric refraction can move the apparent horizon and varies with conditions.
Why can height be zero?
A point on an ideal smooth surface has its geometric horizon at zero distance in this model.