Use this Earth Curvature Calculator to answer two practical questions in seconds: how far you can see from a given eye height, and how much of a distant object is hidden by Earth’s curvature at a given range. Toggle between “Horizon distance” and “Obscured part” to match your task, choose units you’re comfortable with, and (optionally) model atmospheric refraction. Results update instantly and remain read-only so your inputs stay clean.
How the calculator works
The calculator models Earth as a sphere with an adjustable radius. By default it uses the mean Earth radius of 6,371 km (3,959 mi). This is a widely accepted global average and it gives consistent results across locations. You can switch to a custom value if your work requires a specific reference ellipsoid or a specialized geodesy convention. For a quick refresher on Earth’s size see NASA’s overview of Solar System sizes.
Refraction settings let you pick “None,” “Standard,” “Strong,” or a custom coefficient. The app converts that choice into an effective Earth radius behind the scenes. You get precise numbers without juggling different formulas.
Two modes, two kinds of answers
1) Horizon distance
Enter your eyesight level (eye height above local ground or sea level). The calculator returns three values:
- Surface distance to the horizon — arc length along Earth’s surface.
- Line-of-sight distance — the straight slant range from your eye to the tangent point.
- Horizon dip angle — how far the true horizon sits below a perfectly level line at your eye.
These come from exact spherical geometry. When you enable refraction we apply the same relations with an effective radius so you do not need a different set of equations.
2) Obscured part at the target
Enter the distance to the object. If you also know your distance to the horizon you can type it. If you leave it blank the calculator computes it from your eye height and refraction setting. The tool then solves for the hidden height — the minimum top height the object needs to peek above your horizon. If you provide the object’s actual height the result splits it into visible and hidden parts and adds a clear verdict such as “Visible (tops above horizon)” or “Not visible.”
Formulas & models we use
Earth & angles
- Mean Earth radius (default):
R = 6,371,000 m. - Horizon line-of-sight (no refraction):
d_LOS(h) = √(h(2R + h)). - Surface distance to horizon (no refraction):
d_surface(h) = R · acos(R/(R + h)). - Horizon dip angle:
dip(h) = acos(R/(R + h))(radians). Multiply by 180/π for degrees.
Atmospheric refraction via effective Earth radius
Near the surface, air density changes with height. Light bends gently downward which lets you see a little farther. A common first-order model replaces R with an effective radius R' = R/(1 − k). The dimensionless coefficient k describes the ray’s curvature relative to Earth’s curvature. Under “standard” conditions the line of sight curves about one-seventh as much as Earth. In practice this means R' ≈ (7/6)·R and the horizon sits farther away. A typical near-ground value used in geodesy is k ≈ 0.13 though real air varies over space and time.
Hidden height (obscured part)
Let D be the surface (geodesic) distance from you to the target and let d₁ be your surface distance to the horizon at eye height h₁ computed with R’. If the target lies beyond your horizon the remaining arc is Δ = max(D − d₁, 0). The minimum top height that must rise above the horizon is
h_needed(Δ) = R' · (sec(Δ/R') − 1)
This inverts the surface-distance-to-horizon relation with the effective radius. It is exact for the constant-k spherical model you select in the UI.
Unit handling
All calculations run internally in SI units for stability. The interface accepts and displays U.S. customary and SI units interchangeably. Choose feet and miles, or meters and kilometers, or nautical miles for marine use. The converter handles the rest.
What atmospheric refraction does
Refraction is the bending of light in air. Density decreases with height so the refractive index also changes with height. Rays therefore curve downward in a stable atmosphere which lets you see a bit beyond the purely geometric horizon. How much extra range you gain depends on the vertical temperature gradient and humidity profile. Calm mornings often boost range. Strong heating can reduce it.
The constant-k model captures that behavior with a single knob. Choose “None” for a lower bound, “Standard” for a good average, “Strong” when an inversion lifts distant features, or your own custom coefficient if you are matching a particular survey specification.
Quick rules of thumb
When you need a ballpark figure in your head use these. They come from the exact line-of-sight formula simplified for small heights and from the 7/6-Earth approximation for refraction.
| Eye height h (ft) | Geometric horizon (mi) ≈ 1.23 · √h |
With standard refraction (mi) ≈ 1.32 · √h |
|---|---|---|
| 1 | 1.23 | 1.32 |
| 5 | 2.75 | 2.95 |
| 6 | 3.01 | 3.23 |
| 10 | 3.89 | 4.17 |
| 100 | 12.30 | 13.20 |
| 1,000 | 38.90 | 41.74 |
Step-by-step examples
Example A — How far can you see from 6 ft?
- Select Horizon distance.
- Enter eyesight level = 6 ft.
- Leave refraction on Standard for a realistic value.
You’ll see the surface distance to the horizon, the line-of-sight distance, and a small dip angle in degrees. Expect a line-of-sight around 4.17 mi under standard refraction.
Example B — What part of a 1,500-ft tower at 55 mi is hidden?
- Switch to Obscured part.
- Enter distance to the object = 55 mi.
- Enter your eyesight level such as 6 ft.
- Leave “distance to horizon” blank so it is computed automatically.
- Toggle I know target height and enter 1,500 ft.
The calculator subtracts your horizon arc from 55 mi then inverts the geometry to find the height needed to clear the horizon. If the tower is taller than that threshold its top will be visible.
Example C — Will a ship’s mast appear first?
Yes. The hull reaches the curvature first so the mast appears first. Enter the range to the ship under “Obscured part,” choose your eye height, and read the hidden height. Compare that number to the mast height to see if it clears the horizon while the deck stays hidden.
Tips for accurate results
- Use realistic refraction. “None” gives a geometric lower bound. “Standard” is a good baseline for clear or neutral days. “Strong” fits shallow inversions over water. A custom value helps when you must match a professional spec.
- Be consistent with distances. The “distance to the object” is a surface distance. On land it is usually the great-circle distance over the terrain rather than a straight line on a map that cuts through hills.
- Measure eye height carefully. A few feet can shift the horizon by miles near the shore. Measure from your eyes down to the waterline or ground level.
- Remember that air changes. Refraction varies through the day and across landscapes. Heat shimmer and sea breezes can change the apparent horizon by a surprising amount.
- Earth is not a perfect sphere. For visual sight lines a spherical model with the mean radius works well. For survey reductions tied to a particular ellipsoid use the custom radius option.
Glossary
- Distance to the horizon (surface)
- Arc length along Earth’s surface from the observer to the tangent point. Exact formula:
R' · acos(R'/(R' + h)). - Line-of-sight to the horizon
- Straight-line distance from the eye to the horizon point:
√(h(2R' + h)). - Horizon dip angle
- Angle between your horizontal line and the line to the horizon:
acos(R'/(R' + h)). - Obscured (hidden) height
- Minimum top height the target needs to appear above the horizon when it lies beyond your horizon:
R' · (sec(Δ/R') − 1)withΔ = max(D − d₁, 0). - Refraction coefficient (k)
- Dimensionless parameter for near-surface refraction. In this model
R' = R/(1 − k). A common value for ordinary conditions is around 0.13 though it varies with weather and terrain.
Further Reading
- A.T. Young — Distance to the Horizon (derivations and discussion of the 7/6-Earth model).
- University of Washington — How far away is the horizon? (PDF with the classical formulas).
- NASA — Earth size (radius 6,371 km).
- Hirt et al. — Monitoring the refraction coefficient in the lower atmosphere (background on near-ground k values).
- Wikipedia — Atmospheric refraction (broad overview).
Frequently asked questions
Does the calculator assume a flat Earth?
No. Every relation uses a sphere with your selected radius. Refraction is handled by using an effective radius R' = R/(1 − k) which is a standard first-order model.
Why do I see both a surface distance and a line-of-sight distance?
They answer different questions. The surface distance tells you how far you would travel along the ground or water to reach the horizon point. The line-of-sight distance is the straight slant range between your eye and the horizon point.
Is “standard refraction” always correct?
It is a convenient average. Real air changes during the day and across terrain. You can select “None,” “Strong,” or a custom coefficient when conditions differ from the standard assumption.
Do mountains or waves change the math?
They change the inputs rather than the geometry. If the target stands on a hill add that elevation to its height. If you are on a cliff use your eye height above sea level. Rough seas can hide low features even when the top clears the geometric horizon.
Which units should I use?
Use what feels natural. The default shows feet for height and miles for distance. You can switch to SI units or nautical miles with one click. The calculator converts everything consistently in the background.
How does the “I know target height” switch help?
When you know the object’s height the calculator reports two numbers: how much is hidden and how much remains visible. It also prints a plain-language verdict so you can understand the result instantly.
Can I compare different refraction settings?
Yes. Change the setting and watch how the horizon and hidden-height numbers shift. A larger k increases the effective radius which pushes the horizon farther away and can reduce the hidden part.
Why this Earth Curvature Calculator stands out
- Two clear modes: horizon distance and obscured height with the exact outputs you need.
- Read-only results: output chips never mingle with inputs which prevents accidental edits.
- Mobile-first design: tap-friendly inputs, compact helper text, and accessible selectors.
- Comprehensive units: U.S. customary, SI, and nautical miles are all available.
- Refraction controls: none, standard, strong, or a custom k.
- Accurate math: exact spherical formulas with an effective radius for refraction.
Bottom line
Planning a shoreline photo. Checking if a mountain should be visible across the basin. Teaching a geometry lesson that sticks. This Earth Curvature Calculator gives you trustworthy answers in seconds. Enter a height. Pick a distance. Choose a refraction setting that matches your day. The physics stays solid and the interface stays out of your way.
