Earth Curvature Calculator
Estimate curvature drop, horizon distance, and target visibility using spherical Earth geometry.
Tip: This tool uses idealized geometry. Real-world visibility can change due to terrain, weather, humidity, and optical effects.
What this curvature of the earth calculator does
This page gives you a practical way to estimate how Earth’s curvature affects long-distance viewing. You enter a distance, your eye height, and an optional target height. The calculator then returns:
- Curvature drop over the selected distance
- Distance to horizon from your viewing height
- Maximum line-of-sight surface distance between observer and target
- Estimated hidden height of a target at that distance
Core formulas behind the calculator
1) Curvature drop
For a spherical Earth with radius R and surface distance d (meters), an arc-based geometric drop from the local tangent is:
drop = R × (1 − cos(d/R))
For smaller distances, the classic approximation is:
drop ≈ d² / (2R)
This approximation leads to the well-known rule of thumb of roughly 8 inches per mile squared without refraction.
2) Horizon distance from height
Given observer height h, horizon distance along the surface is:
d_horizon = R × arccos(R / (R + h))
The same equation is used for the target height; both horizon distances are added to estimate when the top of the target can become visible.
How atmospheric refraction changes results
Light bends slightly in the atmosphere, typically extending visible range in standard conditions. A common engineering shortcut is to use an effective Earth radius of 7/6 × R. If you check the refraction box, this calculator applies that model. The practical effect is:
- Less apparent drop over a given distance
- Longer horizon distance
- Slightly improved long-range visibility estimates
Example scenarios
Coastal observation
If you stand at about 2 meters eye height and look toward a distant shoreline, you may find that ground-level details disappear first while towers and hills remain visible. This is exactly what the hidden-height estimate captures.
Marine navigation
Boaters often estimate how far away they can detect another vessel or lighthouse. Input your bridge/eye height and the mast or beacon height to estimate likely line-of-sight range.
Surveying and engineering intuition
Over very long alignments, curvature becomes non-trivial. This tool helps build quick intuition before doing project-specific geodetic corrections.
Interpreting the output correctly
- Curvature drop is geometric, not terrain-aware.
- Hidden height is a first-order visibility estimate.
- Visibility can still be blocked by hills, buildings, haze, and Earth not being a perfect sphere.
- The refraction option is a standard approximation, not a guarantee for any specific weather profile.
FAQ
Is this the same as geodetic surveying software?
No. This is an educational and practical estimator based on spherical geometry. Precision surveying uses ellipsoids, datum transformations, and instrument corrections.
Why does target height matter?
Even when Earth blocks the lower portion of a distant object, the upper part can still be visible if it rises above the curvature-hidden section.
Why are my real observations different?
Refraction varies with temperature gradients and pressure, and terrain can dominate visibility outcomes. Use this calculator as a baseline, then adjust for conditions.