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Distance & area

Distance to the horizon calculator

Enter your eye height and see how far you can look before the curve of the Earth hides everything, with the horizon circle on the map. Check whether a distant ship, lighthouse or mountain is visible, and how much of it is hidden.

Standing at the shoreline, about 5.6 ft. In a building, the height of the floor.

Where you’re looking from (for the map)
Can you see something far away?

Example From the 86th-floor deck of the Empire State Building. Change the height or tap the map to look from somewhere else.

Loading the map…

Horizon with refraction Without refraction

Result

The horizon is

42.5 mi

away over the ground, looking from 1,050 ft above sea level, with normal atmospheric refraction

Without refraction
39.7 mi
In a straight line (line of sight)
42.5 mi
Your gaze drops to the horizon by
0.54°

Air bends light slightly downward, letting you see a bit farther. The tool uses the standard refraction coefficient, k = 0.13; strong heat or cold near the ground can change it a lot.

The distant object

Only the top is visible

Hidden by the curve
32 ft71 ft without refraction
Visible part
34 ft
Its top is visible up to
53.2 mi
Diagram: anything below the line of sight is hidden by the curve of the EarthYouObject

Diagram, not to scale: the hidden part is gray.

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How to use it

  1. Enter the height

    Your eye height above the ground or the sea. On a hill or in a building, pick the spot and tick “Add the ground elevation”.

  2. See the horizon

    You get the distance over the ground with normal atmospheric refraction and without it, plus the circle on the map.

  3. Try something far away

    Enter an object’s height and distance: you’ll see whether it’s fully visible, only the top or not at all, and how much the curve hides.

What it’s for

  • Watching the sea

    Find out how far away the ships peeking over the horizon are, from the beach or from a cliff.

  • Hikes and viewpoints

    From a summit, the horizon circle shows which islands, ranges or cities could fit in the view.

  • Photography

    Plan shots of distant mountains or a sunset behind an island.

  • Seeing that the Earth is round

    A ship sailing away disappears from the bottom up: the tool tells you how much of it is hidden.

How it’s calculated

The tool treats the Earth as a sphere with a radius of 6,371 km (3,959 miles). From a height h, your line of sight touches the surface at the horizon; with the radius R it forms a right triangle, which gives the distance.

Air is denser low down than higher up, so light traveling almost level bends slightly toward the ground and you can see a bit farther. That’s modeled as if the Earth were bigger: an effective radius R' = R / (1 − k). With the standard refraction coefficient, k = 0.13 (the average Gauss measured near Hannover, still used in surveying), the effective radius is 4,550 mi (7,323 km) and the horizon is 1.07 times farther away.

Horizon (over the ground)

d = R' · acos(R' / (R' + h)) ≈ √(2R'h)

Without refraction, R' = R = 6,371 km. With refraction, R' = R / (1 − k).

Height hidden by the curve

hidden = R' · (1 / cos((D − d) / R') − 1)

D is the ground distance to the object and d the distance to the horizon. If D is less than d, nothing is hidden.

Examples worked out with the tool
From the shoreline, eyes at 5.6 ft (1.7 m) 3.10 mi (4.99 km); 2.89 mi (4.65 km) without refraction
From the Empire State Building’s 86th-floor deck, at 1,050 ft (320 m) 42.5 mi (68.5 km); 39.7 mi (63.9 km) without refraction
How far your gaze drops to the horizon from the Empire State Building’s 86th-floor deck 0.54°
A ship 66 ft (20 m) tall, 10.0 mi (16.1 km) offshore 27.6 ft (8.4 m) hidden; 34 ft (10 m) without refraction
The Empire State Building’s antenna tip, 1,454 ft (443 m) tall, seen from a beach 40.0 mi (64.4 km) away 790 ft (241 m) hidden; the top 664 ft (202 m) still shows

The 86th-floor open-air deck is 1,050 ft above the street, according to the building; the street itself is a little above sea level, which barely changes the result.

How far you can see from each height

From 6 ft: 3.22 mi (5.18 km), or 3.00 mi (4.83 km) without refraction. From 30 ft: 7.19 mi (11.6 km), or 6.71 mi (10.8 km) without refraction. From 150 ft: 16.1 mi (25.9 km), or 15.0 mi (24.1 km) without refraction. From 300 ft: 22.7 mi (36.6 km), or 21.2 mi (34.1 km) without refraction. From 1,500 ft: 50.8 mi (81.8 km), or 47.4 mi (76.3 km) without refraction. From 3,000 ft: 71.9 mi (116 km), or 67.1 mi (108 km) without refraction. From 10,000 ft: 131 mi (211 km), or 122 mi (197 km) without refraction. The distance grows with the square root of the height: to see twice as far you have to climb four times higher.

Why refraction varies so much

The value k = 0.13 is an average for sight lines well above the ground. Close to a very hot or very cold surface, light can bend much more, or even upward: in measurements 1.8 m (6 ft) above grass on sunny days, k ranged from −4 to +16 (Hirt and others, 2010). That’s why you sometimes see things that should be hidden (or mirages over the sea) and other times see less. Treat the numbers as typical, not as an exact limit.

Frequently asked questions

How far is the horizon from the beach?

With your eyes 5.6 ft (1.7 m) above the water, about 3.10 mi (4.99 km) with normal refraction, or 2.89 mi (4.65 km) without it. Climb onto a dune or a boardwalk a few feet higher and the horizon moves out noticeably.

What is atmospheric refraction?

It’s the bending of light as it passes through layers of air of different density. Near the ground the air is denser, so light curves slightly downward, following the curve of the Earth a little, and lets you see a bit beyond the geometric horizon.

Why add the ground elevation?

Because what matters is your height above sea level, the surface that curves away. On a mountain, adding its elevation gives the horizon over the sea or over a plain at sea level. Other mountains in front of you can block the view sooner.

Does it account for hills in between?

No: the horizon is worked out over a smooth Earth, like the open sea. Hills and buildings in front can block the view sooner, and a distant mountain can be visible beyond the horizon because its peak sticks up.

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