How this calculator works
Tangent drop measures how far a spherical surface falls below the tangent at the starting point. Hidden height is a different quantity: the minimum height of a vertical object that reaches the observer tangent sightline. Observer elevation pushes the horizon outward. Distance here is a surface arc, not a map chord or straight sightline. The model omits terrain, refraction, waves and the non-spherical shape of Earth.
Formula
φ = s/R; α = acos[R/(R+h)]; tangent drop = R(1-cosφ). Hidden height = R[sec(max(0, φ-α))-1].
Worked example
On a sphere of radius 1000 m, a surface arc of 1047.198 m spans 60 degrees. At zero observer height, tangent drop is 500 m but the hidden object height is 1000 m.
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Frequently asked questions
Why are drop and hidden height different?
Drop uses a tangent at the starting surface point. Hidden height uses the observer sightline and the radial direction at the distant object.
What distance should I enter?
Use surface arc distance along the sphere between observer and object.
Can this predict visibility over water?
It gives a geometric baseline. Refraction and variable water level can materially change a real observation.