Distance to Horizon Calculator

Distance to Horizon Calculator for how far you can see from any height. Add refraction, get horizon dip and nautical miles, or find how much of a far object hides behind the curve.

How the Distance to Horizon Calculator Works

The Distance to Horizon Calculator has three modes. Horizon from Eye Height is the default, and it tells you how far you can see. Eye Height for a Distance works backward. Far Object Hidden Height shows how much of a distant ship, island or tower sits behind the curve.

Enter your eye height above the sea or level ground in meters, feet, kilometers, miles, centimeters or inches. Use your eye level, not your standing height. A 1.80 m adult has eyes roughly 10 to 12 cm lower than the top of the head.

The folded Planet, Refraction and Units section picks the world, the refraction model and the units for the results. Earth with standard refraction is the default. The Moon, the planets, Pluto, the Sun and a custom radius are also listed.

Reading the Results

For the default eye height of 1.75 m, the Distance to Horizon Calculator returns 5.10 km. That is the straight sight line from your eye to the horizon, with standard refraction bending the light a little over the curve.

The first card gives the pure geometric horizon, 4.72 km, so refraction adds about 8%. The second card gives the horizon dip, 2.36′ below true eye level. The third card shows the visible surface inside that circle, about 81.7 km².

The fourth card gives the sea horizon from Bowditch’s navigation rule, 2.74 nautical miles. It also shows that a 7.00 m eye height doubles the range. Seeing twice as far always takes four times the height.

Distance to Horizon Formula

The horizon distance comes from a right triangle. Your line of sight just touches the Earth at the horizon, so it meets the radius there at 90°. The hypotenuse runs from the Earth’s center to your eye, R + h.

$$d = \sqrt{h \left( 2R + h \right)}$$

Here d is the distance to the horizon, h is your eye height and R is the Earth’s radius, 6,371 km on average. Both h and R must be in the same unit. This is the exact form the calculator uses.

For any height a person can stand at, h is tiny next to 2R. The formula then shrinks to the version most physics books give, which is accurate to a fraction of a percent below airliner height.

$$d \approx \sqrt{2 R h}$$

Quick Rules in Metric and Feet

Putting R = 6,371 km into √(2Rh) gives a short rule. The distance in kilometers is about 3.57 times the square root of the height in meters. At 10 m that gives 11.3 km, and at 400 m it gives 71.4 km.

In US units, the distance in miles is about 1.22 times the square root of the height in feet, often rounded to 1.23. A 100 ft eye height gives about 12.2 miles of geometric horizon.

With standard refraction the metric factor rises to about 3.86. The Distance to Horizon Calculator shows the quick rule for your settings on its fourth card, so you can check it in your head later.

Atmospheric Refraction

Air is denser near the ground, so light bends gently downward as it travels. That bending carries your sight line a little way past the geometric horizon. The usual way to model it is to swap R for a larger effective radius, k × R.

The calculator uses k = 7/6 as its standard optical setting, which adds about 8% to the geometric distance. Radio and radar engineers use k = 4/3, adding about 15%. Choose None for the pure geometric answer.

Real refraction changes with the weather, especially over water where warm and cold air layers form. On some days a coast normally below the horizon rises into view. On others the horizon sits closer than the table says.

Distance to Horizon Table

The tables below use the same formula as the Distance to Horizon Calculator. They give the geometric horizon and the horizon with standard refraction for common eye heights. Inside the tool, the table also marks the row for your own height.

Choose feet as the height unit and the tool’s table switches to US heights. The table below shows those values in miles, together with Bowditch’s sea horizon in nautical miles from his feet rule, 1.17√h.

Distance to Horizon at 100 Feet and 2,000 Feet

From 100 feet, about the top of a ten-story building, the geometric horizon is 12.25 miles. Standard refraction stretches it to 13.23 miles. Over the sea that is 11.7 nautical miles by Bowditch’s rule.

From 2,000 feet, such as a high hill or a light aircraft, the geometric horizon is 54.8 miles and about 59.2 miles with refraction. At that height the visible circle covers roughly 11,000 square miles of ground.

How Far Is the Horizon at Sea?

At sea the horizon is a clean line, so navigators measure it carefully. The American Practical Navigator, known as Bowditch, gives the distance in nautical miles as 1.17 times the square root of the eye height in feet. In metric it uses 2.07 times the square root of the height in meters.

$$d_{NM} = 2.07 \sqrt{h_m}$$

Standing in a dinghy with eyes about 2 m up, the sea horizon is 2.9 NM away. From a 12 m bridge it is 7.2 NM, and from 25 m it is 10.4 NM. The Distance to Horizon Calculator gives these figures when you set the result unit to nautical miles.

Use the second card for sextant work. Every altitude measured from the sea horizon is corrected for dip. The tool shows the dip from its refraction model next to the almanac rule, 1.76′√h for meters or 0.97′√h for feet.

The Distance to Horizon Calculator picks the Bowditch rule that matches your height unit. Enter 6 ft and it uses 1.17√h, giving a 2.87 NM sea horizon and a 2.38′ dip. The same switch applies to the eye-height and hidden-height modes.

When you range a ship or buoy with a mil reticle, the horizon distance tells you how far you can expect to see its waterline. The Binoculars Range Calculator takes over from there with the target’s size and its mil reading.

Glare off the sea also hides what sits near the horizon. Polarized sunglasses cut that glare best near Brewster’s angle, which the Brewster’s Angle Calculator puts at 53° from vertical for water.

Seeing a Lighthouse or Ship

A tall object has its own horizon. You can see its top as long as your horizon and its horizon together reach across the gap. Navigators call this the geographic range of a light.

Take a light 40 m above the sea, seen from an eye 3 m up. Its geographic range is 2.07 × (√40 + √3) = 16.7 NM. The Distance to Horizon Calculator shows this sum as its maximum sighting range in the Far Object Hidden Height mode.

Visible Horizon and Hidden Height

The Far Object Hidden Height mode works as a visible horizon calculator for objects beyond your own horizon. Enter your eye height, the object’s height and its distance along the surface. The tool returns how much of it is hidden behind the curve.

The default case is a 1.75 m eye and a 30 m object 20 km away. Your horizon is about 5.1 km off, so the object’s lower 14.9 m sit below the horizon line with standard refraction. Without refraction, 18.3 m would be hidden.

The combined horizons reach 26.2 km. Past that distance even the top of a 30 m object drops from view. This is why ships seem to sink from the bottom up as they sail away, while their masts stay in view the longest.

Tall buildings show the same effect on a larger scale. Picture a 442 m skyscraper 80 km across a lake from a 1.75 m eye at the shore. With standard refraction, about 377 m of its height is hidden. Only the top part rises above the water.

How much detail you can make out on that distant top is a separate limit. The Angular Resolution Calculator gives the finest detail a telescope or camera lens can resolve at that range.

Horizon Distance on Other Planets

Smaller worlds curve away faster, so their horizons are closer. The Distance to Horizon Calculator uses each body’s mean radius from NASA’s planetary fact sheets. It treats them as airless spheres, so the answer is the geometric horizon.

An astronaut on the Moon sees the horizon at about half the distance it would be on Earth. Jupiter has no solid surface, so its figure is measured above the cloud level where the pressure is one bar.

How Far You Can See: Quick Answers

These answers cover the horizon questions people search for most, from a beach walk to an airliner window. Each one uses the same formulas as the Distance to Horizon Calculator.

How far is the horizon from a beach?

For an adult standing at the water’s edge, with eyes about 1.7 m up, the horizon is about 5 km or 3 miles away. That is with normal refraction. Sitting on the sand, with eyes about 0.8 m up, brings it in to roughly 3.5 km.

Waves, haze and the lighting usually set a closer practical limit than the horizon itself. On a clear day, though, the distance in the table is where the sea meets the sky.

How far can you see from an airplane?

At a cruising height of 35,000 ft the geometric horizon is about 229 miles. A constant refraction model would add more. At that height most of the light path runs through thin air, so the calculator warns that the geometric figure is the better guide.

The ground you can see from that height covers a circle around 460 miles across. With feet selected, the tool’s height table includes a 35,000 ft row, so you can read this without typing it in.

Why can I see mountains beyond the horizon?

A mountain has its own horizon. A 400 m peak can be seen from sea level up to about 71 km away by geometry, and further with refraction. Add your own horizon and the range grows again.

The Far Object Hidden Height mode of the Distance to Horizon Calculator handles this case. Enter the peak’s height and distance, and it shows how much of the mountain stands above your horizon line.

Does the horizon distance change with latitude?

Only slightly. The Earth is a little flattened, so its radius of curvature changes with latitude and direction. The calculator uses the 6,371 km mean radius, which keeps the answer within about half a percent for any place on Earth.

Weather has a much larger effect than latitude. A strong temperature layer over cold water can shift the visible horizon by several percent from one hour to the next.

Is there a distance to horizon calculator app?

You do not need to install one. The Distance to Horizon Calculator runs in any phone or desktop browser. Use the Share or Print buttons to save a result for a voyage plan or a photo trip.

On a boat, set the result unit to nautical miles and the height unit to feet or meters. The table then lists the sea horizon for every common eye height on deck.