All calculatorsEarth Curvature Calculator
Estimate Earth curvature drop, hidden height, and horizon distance — with optional atmospheric refraction (k-factor), clean unit conversion, step-by-step, and a mini curvature visual. Uses a spherical Earth model (radius R ≈ 6,371,000 m).
Background
Over long distances, the Earth’s surface curves away. That can make distant objects appear “lower” or partially hidden. This calculator uses two common ideas: drop (how far the surface falls below a tangent) and horizon (how far you can see from a given height). You can also enable refraction, which slightly bends light downward and increases visible distance.
How to use this calculator
- Pick a mode (drop, hidden height, horizon, or required height).
- Enter the needed values (distance and/or heights). Choose any units.
- Optionally enable refraction (k-factor) to model typical atmospheric bending.
- Click Calculate to get results, steps, and a mini visual.
How this calculator works
- Curvature drop uses a spherical Earth model.
- For small-to-moderate distances, a great approximation is drop ≈ d² / (2R') where R' is effective radius.
- Horizon distance uses d ≈ √(2R'·h).
- Hidden height
Formula & Equation Used
Earth radius: R ≈ 6,371,000 m
Effective radius (refraction): R' = R / (1 − k)
Drop (approx): drop ≈ d² / (2R')
Horizon distance: d ≈ √(2R'·h)
Dip angle (approx): α ≈ √(2h/R') (radians)
Example Problem & Step-by-Step Solution
Example 1 — Curvature drop over 10 km
- Distance: d = 10,000 m
- Use drop ≈ d²/(2R)
- drop ≈ (10,000²)/(2·6,371,000) ≈ 7.85 m
Example 2 — Horizon from 1.7 m height
- Height: h = 1.7 m
- Use d ≈ √(2Rh)
- d ≈ √(2·6,371,000·1.7) ≈ 4,650 m ≈ 4.65 km
Frequently Asked Questions
Q: What is the k-factor?
It’s a simple way to model atmospheric refraction by increasing the Earth’s effective radius: R' = R/(1−k). Typical values are around 0.13, but conditions vary.
Q: Why do I see “drop” and “hidden height”?
Drop is the surface fall below a tangent at the observer. Hidden height estimates how much of a distant object is below the line-of-sight (tangent) given your observer height.
Q: Is this exact?
It’s a spherical model with a common refraction approximation. Real-world visibility also depends on terrain, temperature gradients, and where exactly you measure heights from (sea level vs ground).