What this calculator does
The horizon is closer than most people expect. At standing eye height of about 1.7 metres it is only around 4.7 kilometres away, which is why a ship on a calm sea disappears so quickly.
Because the relationship runs on a square root, gaining height helps but with diminishing returns. Climbing to ten metres roughly doubles the distance, and reaching a hundred metres only doubles it again.
The formula
The line of sight is tangent to the sphere, forming a right triangle with the planet radius. The distance is the square root of the observer's distance from the centre squared minus the radius squared.
| Term | Meaning |
|---|---|
| Horizon | The furthest point visible before the surface curves away below the line of sight. |
| Tangent line | The line of sight that just grazes the surface, which is what forms the right triangle. |
| Refraction | Atmospheric bending of light, which in reality extends the horizon slightly beyond the geometric figure. |
The inputs explained
| Field | What to enter |
|---|---|
| Observer height above surface (m) | The observer's height above the surface, in metres. Standing eye height is roughly 1.7 m. |
| Planet radius (Earth ≈ 6371000) (m) | The planet radius in metres. Earth averages about 6,371,000. |
When to use it
Working out visibility at sea
How far a lighthouse or vessel can be seen depends on the heights of both the object and the observer.
Planning a radio or microwave link
Line-of-sight communication is limited by the same geometry, which is why masts are built tall.
Understanding the view from altitude
The horizon from a cruising aircraft is hundreds of kilometres away, which puts the scale of the view in perspective.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
How far is the horizon from different heights?
Observer heights from standing to cruising altitude.
| Height above surface | Distance to horizon (km) | Distance in miles |
|---|---|---|
| 1.7 m | 4.654 km | 2.89 mi |
| 10 m | 11.288 km | 7.01 mi |
| 100 m | 35.696 km | 22.18 mi |
| 10,000 m | 357.099 km | 221.89 mi |
Questions
Why is the horizon so close?
Because Earth is large but the observer is very low on it. At 1.7 metres you are looking along a tangent that grazes the surface within a few kilometres, even though the planet is thousands of kilometres across.
Does atmospheric refraction change this?
Yes, and this calculation ignores it. Air bends light slightly downward, following the curve, which typically extends the visible horizon by around 8 per cent beyond the geometric figure.
How far can I see a tall object?
Further than the horizon distance, because the object has its own horizon. Add the two horizon distances together: a 30 metre lighthouse is visible from much further than your own horizon alone would suggest.
Why does doubling height not double the distance?
Because the relationship is essentially a square root of height. To double the horizon distance you need roughly four times the height, which is why observation towers give diminishing returns.
For how far a curve drops below the line of sight, see the Earth curvature drop calculator. For arc geometry generally, see the arc length calculator.