How the Distance to the Horizon is Calculated
The horizon is the point where the Earth's curved surface drops below your line of sight. The higher you are, the further away that point is, because you can see further around the curve of the planet before it falls away. This calculator uses the standard geometric formula for horizon distance, based on your height above the surface and the radius of the Earth.
The Geometric Formula
The basic geometric distance to the horizon is:
d = √(2 × R × h)
Where d is the distance to the horizon, R is the radius of the Earth (approximately 6,371 km), and h is your height above the surface, in the same units as R. This formula comes from a simple right-angle triangle: your eye, the centre of the Earth, and the point on the horizon form a triangle where the line of sight is tangent to the Earth's surface. Because h is tiny compared to R, the formula simplifies to d approximately equal to the square root of 2Rh.
Correcting for Atmospheric Refraction
Light does not travel in a perfectly straight line through the atmosphere. Air density decreases with altitude, which bends (refracts) light slightly downward as it travels long distances close to the surface. This effect lets you see a little further than the pure geometric formula predicts. Surveyors and navigators commonly account for this using an effective Earth radius about 7/6 (roughly 1.07 times) the true radius, giving a refraction-corrected horizon distance a few percent greater than the simple geometric result. The exact amount of refraction varies with temperature, pressure and humidity, so this correction is a widely used average, not an exact value for every condition.
Worked Example
A person standing at the beach with their eyes 1.7 m above sea level has a geometric horizon distance of about 4.65 km. With the standard refraction correction (an effective Earth radius of 7/6 the true radius), the visible horizon extends to about 5.03 km, roughly 370 m further. This is why on a clear day at the beach, the horizon looks close, but climbing even a small dune or headland noticeably pushes it further away.
Why Height Matters More Than You Would Think
Because the formula depends on the square root of height, small increases near ground level make a noticeable difference, while very large increases in height give diminishing returns. Doubling your height from 1.7 m to 3.4 m increases the horizon distance by about 41 percent, not 100 percent. This is why lighthouses and observation towers are built tall: even a modest increase in height meaningfully extends the visible horizon, but each extra metre above that gives a smaller improvement.
Common Uses
- Estimating how far out to sea a lighthouse or lookout can be seen from
- Working out line-of-sight range for radio, radar or visual signalling
- Understanding why ships appear to "sink" below the horizon as they sail away
- Photography and surveying, when working out visible range from elevated positions
- General curiosity about how far you can actually see from a given height
Related Calculators
Sources: Standard geometric horizon distance formula (d = √(2Rh)), used widely in navigation, surveying and optics references. Earth mean radius of 6,371 km (International Union of Geodesy and Geophysics). Atmospheric refraction correction factor of approximately 7/6, as commonly applied in surveying and marine navigation texts.
This calculator gives an estimate based on a spherical Earth model and an average atmospheric refraction correction. Actual visible distance can vary with atmospheric conditions, terrain, obstructions and the height and size of the object you are trying to see, and does not account for Earth's slight oblate shape.
