How far away is the horizon?

The geometry

Draw a line from your eye that just touches the Earth. That line meets the radius at a right angle, so the eye, the tangent point and the Earth's center form a right triangle. The hypotenuse is R + h, one side is R, and the other side is the distance d to the horizon. Pythagoras gives d² = (R + h)² - R² = h · (2R + h).

Because h is tiny next to 2R for anyone not in orbit, the formula simplifies to d ≈ √(2Rh). With R = 6,371.0088 km and h in meters, d in kilometers is about 3.57 · √h. In miles with h in feet, d is about 1.22 · √h.

Horizon distance by height

The viewpoint labels are round example heights, not measurements of any specific building or peak. Plug in the real height of the place you care about in the calculator below.

Refraction lets you see a little farther

Air bends light slightly downward because it is denser near the ground. That bend carries your line of sight a little past the geometric horizon. How much depends on the temperature profile of the air, which changes by the hour, so no single number is exact. The calculator on this site offers a standard adjustment that multiplies the geometric distance by 1.0746. Over cold water on a warm day the real value can be larger or smaller, and mirages are the extreme case.

Seeing a distant object over the horizon

A lighthouse or a mountain has its own horizon. You can see its top when the two horizon distances add up to more than the gap between you. From 1.7 meters your horizon is 4.7 km; a 100 meter tower's horizon is 35.7 km. So the top of that tower is visible out to about 40.4 km across flat water, ignoring haze and refraction.

Over land, terrain usually blocks the view long before the curvature does. Use the elevation profile to check whether a ridge sits between two points.

How far the horizon dips

From height h the horizon appears below eye level by a small angle, the dip, equal to atan(d / R). At 1.7 meters the dip is about 2.5 arc minutes; from 11,000 meters it is about 3.4 degrees. Celestial navigators correct sextant readings for this dip.

Worked example: a 30 meter viewpoint

Take an eye height of 30 meters, roughly a ten story building. Convert to kilometers, then apply the exact formula:

h = 30 m = 0.030 km
2R + h = 2 × 6,371.0088 + 0.030 = 12,742.0476 km
h · (2R + h) = 382.2614 km²
d = √382.2614 = 19.55 km = 12.15 mi

The shortcut 3.57 × √30 gives 19.55 km, close enough for any real use. Keep the units straight: the shortcut takes meters and returns kilometers, and the miles version takes feet and returns miles.

Horizon distance is not visibility

The horizon distance is how far the surface is visible before it curves away. Whether you can make out anything there depends on haze, humidity, light and contrast. On many days the air limits the view long before the curve does. Treat the numbers here as the upper bound for a clear day over flat water or a flat plain.

Using the horizon calculator

Enter an eye height in feet or meters in the calculator below. It returns the geometric horizon, the horizon with the standard refraction adjustment, and the dip angle. For a viewpoint on a hill, add the hill's elevation above the surface you are looking across to your eye height. The elevation finder gives ground elevation for any point.