Hong Life CalculatorsUnderstanding the distance to the horizon calculator — concept and principle
How far can you see before the Earth's curve hides the surface? This calculator finds the distance to the horizon from your eye height, with or without atmospheric refraction, the dip of the horizon, and how far away a tall object can still be seen.
How it is calculated
- Geometric horizon: d = √(2Rh + h²) ≈ √(2Rh), with R = 6,371 km (NASA mean Earth radius).
- With standard refraction, an effective radius R ÷ (1 − k) is used, with k ≈ 0.13. This gives the navigators' rule d (nmi) ≈ 2.07 × √h (m), or about 3.83 km × √h.
- Two objects can see each other while the sum of their horizon distances exceeds the distance between them.
- Dip = how far the horizon lies below eye level.
The principle
Your line of sight to the horizon is a tangent to the sphere. By Pythagoras, (R + h)² = R² + d², which gives the formula. Because h is tiny compared with R, the h² term hardly matters. Air density decreases with height, so light bends slightly downwards, following the Earth's curve a little and letting you see farther than pure geometry predicts.
Worked example
Default inputs: eye height 1.7 m, standard refraction (k = 0.13), no distant object.
- Effective radius = 6,371 ÷ 0.87 ≈ 7,323 km.
- d = √(2 × 7,323,000 × 1.7) ≈ 4,990 m = 4.99 km.
- In other units: 2.69 nautical miles, 3.1 miles.
- Dip of the horizon ≈ 2.3 arcminutes.
- No object height was entered, so the farthest sighting distance shows —.
| Eye height above the water or flat ground (m) | Distance to the horizon (km) |
|---|---|
| 0.425 | 2.49 |
| 0.85 | 3.53 |
| 1.27 | 4.32 |
| 1.7 | 4.99 |
| 2.12 | 5.58 |
| 2.55 | 6.11 |
| 3.4 | 7.06 |
How to read the result
A person standing on a beach sees the sea horizon about 5 km away. From a 100 m cliff it is roughly 38 km. Lighthouses are visible much farther because their own height adds its horizon distance to yours. Real refraction varies with weather, especially over cold water or hot ground, so distances can differ noticeably.
Common mistakes
- Using height in feet with a metric formula.
- Forgetting to add the object's own horizon distance when asking if it is visible.
- Assuming refraction is always the standard value.
- Applying the formula over hills, which block the view first.
Frequently asked questions
How far is the horizon at sea level?
For eye height 1.7 m, about 4.7 km geometrically and about 5 km with standard refraction.
How far can I see from 100 m up?
About 38 km with standard refraction.
Why does refraction increase the distance?
Light bends slightly downwards in the atmosphere, following the Earth's curvature.
Sources: NASA, Earth Fact Sheet (mean radius 6,371 km) · National Geospatial-Intelligence Agency, The American Practical Navigator (Bowditch), Pub. No. 9 (distance to the horizon)