How Far Is the Horizon? Distance by Eye Height, With the Formula

For a person whose eyes are 5 ft 6 in (1.68 m) above the sea, the horizon is about 2.9 miles (4.6 km) away by pure geometry and about 3.1 miles (5.0 km) with normal atmospheric refraction. A 6-foot eye height gives 3.0 and 3.2 miles. That is the answer for a flat sea or a large, flat plain; on land, hills and buildings usually block the view much sooner.

Height changes the distance slowly, because it grows with the square root of eye height. To double how far you can see, you have to climb four times as high:

• Lying on the beach, eyes about 1 ft up: about 1.3 miles.
• Standing at the waterline: about 3 miles.
• From a 100-foot cliff: about 13 miles.
• From 1,000 feet: about 42 miles.

The horizon distance calculator runs the same formula for any height, in feet or meters, and checks whether a distant object of a given height would show above the horizon.

The horizon is where your line of sight just grazes the Earth's surface. At that point the sight line is tangent to the surface, so it meets the Earth's radius at a right angle. That right triangle (your eye, the horizon point, the Earth's center) gives everything:

(R + h)² = R² + d², so d = √(2Rh + h²)

Here R is the Earth's radius, h your eye height and d the straight-line distance to the horizon. Because h is tiny compared with R (even at 35,000 feet it is 0.17% of R), the h² term can be dropped: d ≈ √(2Rh).

With the mean radius R = 3,959 miles (6,371 km) (the Earth is slightly flattened, as explained in how many miles around the Earth, but a mean radius is accurate to about half a percent here), the constants become:

Worked example: eyes 6 ft above the water. √6 = 2.45. Geometric distance: 1.22 × 2.45 = 3.0 miles. With refraction: 1.32 × 2.45 = 3.2 miles. In metric, 6 ft is 1.83 m; √1.83 = 1.35, and 3.86 × 1.35 = 5.2 km.

The distance d is measured along the sight line, but for any height below a few miles it is practically the same as the distance along the curved surface.

Computed with d = √(2Rh + h²) and R = 6,371 km, then again with the effective radius of 7/6 R used for standard refraction:

Heights for the Burj Khalifa (828 m tall, deck at 555 m) and Everest (8,848.86 m, the 2020 Nepal-China survey) are the published figures; we ignore the few meters of ground elevation under the tower. The Everest row assumes a sea-level horizon; in reality, the surrounding Himalaya and the Tibetan Plateau, thousands of meters high, block and shorten the view.

The chart makes the square-root law visible: the first 100 feet of height buy 13 miles of view, the next 900 feet only 29 more.

The horizon is not the limit of what you can see. A tall object beyond your horizon can still poke above it, because the object has its own horizon. The two distances add:

Maximum visible distance = d(your eye height) + d(object height)

Examples with standard refraction and a 6-ft eye height (3.2 mi):

• A 100-ft ship's mast or a 100-ft bluff: 3.2 + 13.2 = 16.5 miles.
• A 150-ft lighthouse: 3.2 + 16.2 = 19.4 miles, which is why lighthouse lights are placed high.
• A 1,000-ft headland or skyscraper: 3.2 + 41.8 = 45 miles.

Before the top disappears, the bottom goes first. The height hidden below your horizon at distance D is about (D − d)² / (2R), where d is your own horizon distance. From a 6-ft eye height:

That is why ships appear "hull down" as they sail away and why a city skyline across a wide lake shows only its upper floors. The horizon distance calculator does both calculations, including the hidden height for a target at a given distance.

Air gets thinner with height, and light bends slightly as it passes through the density gradient, curving downward to follow the Earth a little. You see a bit over the geometric horizon. Atmospheric scientist Andrew T. Young describes the usual approximation: the light path curves at about one seventh of the Earth's curvature, which is the same as replacing R with an effective radius of 7/6 R. That adds about 8% to the distance (the square root of 7/6 is 1.08).

The 7/6 rule describes a typical day, not every day:

• Temperature inversions, common over cold water or on calm mornings, bend light more strongly. Distant coasts can "loom" into view and the horizon moves farther out.
• Hot surfaces such as summer pavement, desert or a sun-heated beach can bend light the other way and pull the horizon in.
• Haze and humidity do not move the horizon, but they often limit what you can actually see long before the geometry does.

Young notes the simple formulas are typically good to a few percent, but mirages can make them wildly off. Different references use slightly different refraction values: our calculator uses a refraction coefficient of k = 0.13, which adds about 7.2%, close to the 7/6 rule. Surveyors and radio engineers use their own conventions; radio waves, for example, are often modeled with a 4/3 R effective radius.

From an airliner at 35,000 feet the horizon is about 230 to 250 miles away, so on a clear day the window shows a circle of Earth roughly 500 miles across. The horizon also sits below eye level. The angle of this "dip" is about 3.3° at cruise altitude, compared with 0.04° for a person on a beach. That is small enough that the horizon still looks almost straight from a passenger window; the curvature becomes obvious only from much higher altitudes, such as high-altitude balloons or spacecraft.

On mountains the geometry is the same, but the horizon is rarely the sea. From Everest, the theoretical sea-level horizon is about 225 miles away, yet the view is cut off by nearby peaks and limited by haze. The best real long views are from isolated high points that overlook lower ground or the sea.

If you are inland, use your height above the surface you are looking across, not your elevation above sea level. Standing on a 5,000-ft plateau that stays flat for miles, your eye height is still 5 to 6 feet. The my elevation tool reports your elevation above sea level, which is the right input only when you overlook the sea or land near sea level. For a viewpoint you plan to visit, look up both the summit and the ground you will look across with the elevation finder and use the difference.

A few errors come up again and again:

• Using elevation instead of height above the view. A hiker at 9,000 ft looking across a valley floor at 6,000 ft has an effective height of about 3,000 ft, not 9,000.
• Forgetting the target's height. "The horizon is 3 miles away" does not mean you cannot see a lighthouse 15 miles away; add both horizon distances.
• Mixing units. The 1.22 constant needs feet in and gives miles out; the 3.57 constant needs meters in and gives kilometers out. Feeding meters into the miles formula understates the result by about 45%.
• Using the "8 inches per mile squared" rule for line of sight. That rule gives the drop of the surface below a level line from the point where you stand: 8 inches at 1 mile, 32 inches at 2 miles, 6 feet at 3 miles. It ignores your eye height and refraction, so it overstates how much of a distant object is hidden.
• Expecting a hard edge. On a calm sea the horizon is sharp, but waves, haze and refraction often blur it by a large fraction of a mile.