Horizon Distance Calculator
Calculate the maximum distance at which an observer can see a target object over the horizon, based on both their heights above sea level. Ideal for navigation, broadcasting tower placement, and outdoor planning.
About this calculator
Because Earth's surface is curved, an observer can only see to a finite distance before the horizon cuts off their line of sight. The geometric horizon distance depends on the square root of the observer's height above the surface. When both observer and target have elevation, the total visible range is the sum of each party's individual horizon distances. The formula is: D = 3.57 × (√observerHeight + √targetHeight), where heights are in metres and distance is in kilometres. The constant 3.57 km/m^0.5 accounts for Earth's mean radius (~6,371 km) and includes a standard atmospheric refraction correction that effectively increases the optical horizon by about 7% compared to pure geometric calculation. This makes the result more accurate under typical atmospheric conditions.
How to use
An observer stands on a cliff 49 m above sea level, trying to spot a lighthouse whose lamp is 16 m above sea level. Step 1 — Square root of observer height: √49 = 7. Step 2 — Square root of target height: √16 = 4. Step 3 — Sum: 7 + 4 = 11. Step 4 — Multiply by 3.57: 11 × 3.57 = 39.27 km. The lighthouse lamp becomes visible at a maximum distance of approximately 39.3 km under standard atmospheric conditions.
Frequently asked questions
How does atmospheric refraction affect the horizon distance calculation?
In a vacuum, the geometric horizon would be slightly closer than what we observe, because light travels in straight lines. In the real atmosphere, light bends slightly downward due to the decrease in air density with altitude, effectively making Earth appear to have a larger radius for optical purposes. The constant 3.57 used in this formula already incorporates a standard refraction correction equivalent to using an effective Earth radius of about 8,500 km instead of 6,371 km. In unusual atmospheric conditions such as temperature inversions, refraction can be much stronger, enabling objects to be seen far beyond the standard horizon.
Why does the target's height matter for calculating how far away you can see it?
The observer's horizon marks the point at which a sea-level object would disappear, but a tall target rises above that geometric line and remains visible from greater distances. Adding the target's own horizon distance (3.57 × √targetHeight) extends the total line-of-sight range. This is why lighthouses and broadcast towers are built tall — a 100 m tower can be seen from about 35.7 km away even by an observer at sea level. Combining the heights of both parties gives the total maximum distance at which they can mutually see each other.
When is the horizon distance calculator useful for navigation and engineering?
Mariners use horizon distance calculations to determine how far away a lighthouse or coastline becomes visible, which is critical for safe passage planning. Radio and TV broadcast engineers use it to estimate the line-of-sight coverage area of a transmitter tower placed at a given elevation. Drone operators need it to comply with visual line-of-sight regulations, which require the aircraft to remain visible to the pilot. Hikers and military observers also use it to assess how far they can observe from a hilltop or ridgeline.