How far away is the horizon?

It is the tangent from your eye to the sphere, sqrt(2Rh + h²) with R the radius and h your eye height. For a standing adult at 1.7 m on a sphere of 6,371 km, that is 4.654 km. Climb to 100 m and it goes out to 35.696 km. The horizon grows with the square root of height, so twice as high is only about 1.4 times as far, which is why a mast helps far less than people expect.

How much of a distant object does the curve hide?

Less than the drop, and this is the mistake almost every page makes. Your sight line is already touching the surface at the horizon, so only the distance beyond the horizon hides anything. At an eye height of 1.7 m, a target 20 km away sits 15.346 km past your horizon, and the height it loses is the drop over that last stretch alone: 18.48 m. The surface has fallen 31.39 m over the whole 20 km, which is a different number answering a different question.

What is the eight inches per mile rule?

It is the tangent drop over the first mile of level surface, and this page computes it as 8.00 in when you enter a mile with heights in feet. It is not a rate you can multiply: the drop grows with the square of the distance, so doubling the distance nearly quadruples the drop rather than doubling it. Applied as eight inches for every mile it overstates short distances and badly understates long ones.

Does this include atmospheric refraction?

Not unless you ask for it. The default figures are geometric. Switching refraction on divides the radius by 1 minus k, the standard model, with k = 0.13 for average low-level air: that moves a 1.7 m horizon from 4.654 km out to 4.990 km and cuts what a 20 km target loses from 18.48 m to 15.38 m. Surveyors often use one seventh instead, and over hot ground the coefficient can fall below zero, so you can also enter your own.

Why do curvature calculators disagree with each other?

Four conventions, usually unstated. Some report the drop and call it the hidden height. Some fold refraction in silently while others do not. Some measure distance along the line of sight and others along the surface. And the radius differs: the Earth is not a sphere, so 6,371 km is a mean, not a measurement of the ground under you. This page states all four beside the result: distance along the surface, heights along the local vertical, refraction off unless you switch it on, and whatever radius is in the field.

Can I use a different planet, or a different radius?

Yes. The radius field takes any radius from one metre to a billion kilometres, in kilometres or miles, and every figure on the page is computed from whatever is in that field. A smaller world curves away faster: the same eye height gives a nearer horizon and hides a distant target sooner. Nothing here is specific to the Earth except the default value.

Is this good enough for sighting or navigation?

No, and it is not built for it. The geometry is exact for the sphere and heights you enter, but the real atmosphere bends light by an amount that changes with temperature, pressure and the gradient over the surface, especially over water and in the first few metres of air. Treat the numbers as a geometry exercise: a photography or long-distance viewing sanity check, not a decision you act on.

Earth Curvature Calculator

Enter an eye height, a distance and the radius of the sphere, and this page returns the three numbers the question usually mixes together: the distance to the horizon, the tangent drop of the surface over that distance, and the height a target at that distance loses behind the curve. The last two are different quantities, and the hidden height is the smaller one, because only the stretch beyond the horizon hides anything. Every figure is exact spherical geometry printed beside the formula that produced it, with a cross-section underneath, and refraction stays off until you switch it on and name a coefficient.

Horizon4.654 kmeye height 1.7 m · R 6,371 km · geometric, refraction off

Eye height

Distance to target

Radius

Refraction

Geometric figures for the sphere you entered. Light travels in straight lines and the air is ignored.

Horizon

4.654 km

Drop at 20 km

31.39 m

Hidden at 20 km

18.48 m

Refraction

Off

Not to scale. A cross-section from the observer on the left to the target 20 km away. The surface curves away, the sight line grazes it at the horizon 4.654 km out, and by the target the sight line stands 18.48 m above the surface: that is the height hidden from view. The surface itself has fallen 31.39 m below the level of the observer's feet.

The working

Distance is measured along the surface and heights along the local vertical. Refraction is off, so every figure is the geometric one for the radius you entered.

Horizon, line of sightsqrt(2Rh + h²), R = 6,371,000 m, h = 1.7 m
4.654 km
Horizon, along the surfaceR × atan(sqrt(2Rh + h²) / R), R = 6,371,000 m
4.654 km
Tangent drop at 20 kmR × (1/cos(d/R) - 1), d = 20 km
31.39 m
Distance beyond the horizond - horizon along the surface = 20 km - 4.654 km
15.346 km
Hidden at 20 kmR × (1/cos(b/R) - 1), b = 15.346 km
18.48 m

Only the 15.346 km beyond the horizon hides anything. The surface drops 31.39 m over the full 20 km, but a target there loses 18.48 m, which is the drop over that last stretch alone.

Worked examples

Earth's mean radius, 6,371 km, with refraction off. Every figure here is computed by the same code as the answer above, so the examples and the tool can never disagree.

Eye height 1.7 mgeometric horizon
4.654 km
Eye height 100 msame radius
35.696 km
Target 20 km away, eye 1.7 mbeyond the horizon
18.48 m hidden below the sight line

Accuracy. Exact spherical geometry on the radius and heights you enter: the horizon is sqrt(2Rh + h²), and tangent drop and hidden height are kept apart on the page because they answer different questions and are often confused. Refraction is off until you switch it on and name a coefficient, and it is then labelled as a model. This is a geometry exercise, not sighting or navigation guidance.

Common questions

How far away is the horizon?: It is the tangent from your eye to the sphere, sqrt(2Rh + h²) with R the radius and h your eye height. For a standing adult at 1.7 m on a sphere of 6,371 km, that is 4.654 km. Climb to 100 m and it goes out to 35.696 km. The horizon grows with the square root of height, so twice as high is only about 1.4 times as far, which is why a mast helps far less than people expect.
How much of a distant object does the curve hide?: Less than the drop, and this is the mistake almost every page makes. Your sight line is already touching the surface at the horizon, so only the distance beyond the horizon hides anything. At an eye height of 1.7 m, a target 20 km away sits 15.346 km past your horizon, and the height it loses is the drop over that last stretch alone: 18.48 m. The surface has fallen 31.39 m over the whole 20 km, which is a different number answering a different question.
What is the eight inches per mile rule?: It is the tangent drop over the first mile of level surface, and this page computes it as 8.00 in when you enter a mile with heights in feet. It is not a rate you can multiply: the drop grows with the square of the distance, so doubling the distance nearly quadruples the drop rather than doubling it. Applied as eight inches for every mile it overstates short distances and badly understates long ones.
Does this include atmospheric refraction?: Not unless you ask for it. The default figures are geometric. Switching refraction on divides the radius by 1 minus k, the standard model, with k = 0.13 for average low-level air: that moves a 1.7 m horizon from 4.654 km out to 4.990 km and cuts what a 20 km target loses from 18.48 m to 15.38 m. Surveyors often use one seventh instead, and over hot ground the coefficient can fall below zero, so you can also enter your own.
Why do curvature calculators disagree with each other?: Four conventions, usually unstated. Some report the drop and call it the hidden height. Some fold refraction in silently while others do not. Some measure distance along the line of sight and others along the surface. And the radius differs: the Earth is not a sphere, so 6,371 km is a mean, not a measurement of the ground under you. This page states all four beside the result: distance along the surface, heights along the local vertical, refraction off unless you switch it on, and whatever radius is in the field.
Can I use a different planet, or a different radius?: Yes. The radius field takes any radius from one metre to a billion kilometres, in kilometres or miles, and every figure on the page is computed from whatever is in that field. A smaller world curves away faster: the same eye height gives a nearer horizon and hides a distant target sooner. Nothing here is specific to the Earth except the default value.
Is this good enough for sighting or navigation?: No, and it is not built for it. The geometry is exact for the sphere and heights you enter, but the real atmosphere bends light by an amount that changes with temperature, pressure and the gradient over the surface, especially over water and in the first few metres of air. Treat the numbers as a geometry exercise: a photography or long-distance viewing sanity check, not a decision you act on.

Exact spherical geometry on the radius and heights you enter: the horizon is sqrt(2Rh + h²), and tangent drop and hidden height are kept apart on the page because they answer different questions and are often confused. Refraction is off until you switch it on and name a coefficient, and it is then labelled as a model. This is a geometry exercise, not sighting or navigation guidance.