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The horizon is about 3 miles away — 2.9 miles, or 4.7 km — when you stand on a beach with your eyes 5 ft 7 in (1.70 m) above the sand. Counting the refraction of the air, which bends light and lets you see slightly farther, it is closer to 3.1 miles (5 km). That is an hour's walk: the edge of the visible world is far nearer than it looks.
The circle above draws those 2.9 miles from Brighton beach. Drag the map to your own beach, window or hilltop to see how far your horizon reaches.
It depends on one thing only: how high your eyes are. With the height h, the distance to the horizon is:
| Formula | What it gives | Eyes at 5 ft 7 in |
|---|---|---|
| d ≈ 1.22 × √h (miles, h in feet) d ≈ 3.57 × √h (km, h in metres) | Geometric horizon (bare Earth, no air) | 2.9 mi / 4.7 km |
| d ≈ 1.32 × √h (miles, h in feet) d ≈ 3.86 × √h (km, h in metres) | Real horizon, with normal atmospheric refraction | 3.1 mi / 5.0 km |
The first comes straight from Pythagoras: the line from your eyes to the point where your sight grazes the ground is tangent to the sphere, so it measures √(2·R·h) with R = 6,371 km, the Earth's radius. Run the numbers and √(2 × 6,371,000) = 3,569 — hence the 3.57. Sailors know the imperial version as √(7h/4).
The second adds the air. The atmosphere is denser at the bottom than at the top, so light rays bend slightly downwards and follow the curve of the planet a little: you see farther than geometry allows. Under normal conditions it works out as an Earth with 7/6 of the real radius, about 8 % more reach (Andrew T. Young, SDSU). Over the sea, temperature inversions can push it much further — which is why coastlines and ships that "should" be hidden sometimes appear as mirages.
| Where you are looking from | Eye height | Horizon (geometric) | With refraction |
|---|---|---|---|
| Lying on your towel | 8 in (0.2 m) | 1.0 mi (1.6 km) | 1.1 mi (1.7 km) |
| Standing on the beach | 5 ft 7 in (1.70 m) | 2.9 mi (4.7 km) | 3.1 mi (5.0 km) |
| Seafront promenade or a dune | 16 ft (5 m) | 5.0 mi (8.0 km) | 5.4 mi (8.6 km) |
| A lookout tower | 33 ft (10 m) | 7.0 mi (11.3 km) | 7.6 mi (12.2 km) |
| A tenth-floor window | 98 ft (30 m) | 12.2 mi (19.6 km) | 13.1 mi (21.1 km) |
| A lighthouse | 164 ft (50 m) | 15.7 mi (25.2 km) | 17.0 mi (27.3 km) |
| A cliff top | 328 ft (100 m) | 22.2 mi (35.7 km) | 24.0 mi (38.6 km) |
| A skyscraper | 820 ft (250 m) | 35.1 mi (56.4 km) | 37.9 mi (61.0 km) |
| A coastal hill | 1,640 ft (500 m) | 49.6 mi (79.8 km) | 53.6 mi (86.3 km) |
| The summit of Ben Nevis | 4,413 ft (1,345 m) | 81.4 mi (130.9 km) | 88.0 mi (141.6 km) |
| A cruising airliner | 32,808 ft (10,000 m) | 221.8 mi (357 km) | 239.8 mi (386 km) |
From the International Space Station, 250 miles (400 km) up, the horizon runs out to about 1,425 miles (2,300 km): half of Europe in one glance. That high the short formula breaks down and you need the full one, √(h² + 2·R·h).
The square root in the formula is the whole story, and it explains what everyone notices on a beach: standing on tiptoe achieves nothing. Because distance grows with the square root of height, doubling your horizon means quadrupling your height above the ground.
Going from eye level (5 ft 7 in) to a second-floor window (22 ft) takes the horizon from 3.1 to 6.3 miles. Doubling it again means climbing to 89 ft, and again, to 357 ft. That is why lighthouses are built tall and on headlands, and why it takes an aircraft to push the horizon out to hundreds of miles.
Because the horizon is not a wall: it is the point where your line of sight grazes the ground. Anything taller than that grazing point pops back into view behind it. To know whether two things can see each other, add their two horizons:
d ≈ 1.32 × (√h₁ + √h₂) in miles and feet, or 3.86 × (√h₁ + √h₂) in kilometres and metres.
Standing on the beach (3.1 mi of horizon) you can see the top of a 1,640 ft hill (53.6 mi of horizon) up to 57 miles (91 km) away — while the village at its foot, at sea level, has been hidden for the last fifty miles.
The extreme case is the world record for the longest photographed line of sight: 443 km (275 miles) from Pic de Finestrelles (2,826 m, in the Pyrenees) to Pic Gaspard (3,883 m, in the Écrins in the Alps), shot by Marc Bret at dawn on 16 July 2016 (Beyond Horizons). Here is the lovely part: pure geometry caps that pair of mountains at 412 km. The photograph should not exist. With refraction the ceiling rises to 446 km. Those 443 km are only possible because the air bends light.
Another way to look at it: how far the Earth's surface falls below your line of sight. The drop is 8 inches × d² with the distance in miles (0.0785 m × d² in kilometres):
| At this distance… | The Earth drops by |
|---|---|
| 1 mile (1.6 km) | 8 in (20 cm) |
| 3 miles (4.8 km) | 6 ft (1.8 m) |
| 6 miles (9.7 km) | 24 ft (7.3 m) |
| 12 miles (19.3 km) | 96 ft (29 m) |
| 30 miles (48 km) | 600 ft (183 m) |
Hence the classic sight of a ship sailing away and disappearing from the bottom up, hull first and mast last. For someone standing on the beach, a yacht with a 33 ft mast is still visible — the mast alone — out to about 10.7 miles (17 km), more than three times as far as your own horizon.
Click any of them to see it above, centred on Brighton:
| Looking from | Horizon |
|---|---|
| The beach (5 ft 7 in) | 2.9 mi (4.7 km) |
| A tenth-floor window (98 ft) | 13.1 mi (21.1 km) |
| A cliff top (328 ft) | 24 mi (38.6 km) |
| Ben Nevis (4,413 ft) | 88 mi (141.6 km) |
| An airliner (32,808 ft) | 240 mi (386 km) |
The circle is a straight-line radius from the centre of the map, so it draws exactly your horizon: everything inside it is, in theory, in view. If what you want is the distance between two specific points — your window and that mountain — use the measure a distance tool.
Want to keep playing with distances? See how long a nautical mile is, how long a marathon is, or open the distances tool and draw your own. And if areas are more your thing, there is the Hectareometer.