Calculator
Formula used
For an arc distance D on a sphere of radius R, the central angle is: θ = D / R (radians).
The curvature “drop” below the start tangent line (sagitta) is: h = R · (1 − cos θ). For short distances, a useful approximation is: h ≈ D² / (2R).
Atmospheric refraction is modeled by an effective radius: Reff = R / (1 − k), where k is the refraction coefficient.
How to use this calculator
- Enter your distance in miles.
- Choose a step size to build the distance table.
- Select an Earth radius preset or choose a custom radius.
- Set refraction coefficient k if needed.
- Press Calculate to view results above the form.
- Use the buttons to download CSV or PDF.
Example data table
Example uses mean radius, k = 0.13, and step = 1 mile.
| Distance (mi) | Approx drop (in) | Approx drop (ft) | Notes |
|---|---|---|---|
| 1 | ~6.9 | ~0.58 | Near-horizon curvature over one mile. |
| 5 | ~172 | ~14.3 | Useful for long flat lake views. |
| 10 | ~688 | ~57.3 | Often cited in surveying discussions. |
| 20 | ~2750 | ~229 | Curvature grows with distance squared. |
Article: Curvature of Earth per mile
Why curvature per mile matters
Curvature becomes noticeable when you compare a straight reference, like a laser level, to the curved surface below it. Surveying, road design, lake photography, and long‑range radio links all use a drop estimate to avoid surprises at the horizon. At 20 miles, the geometric drop is roughly 229 feet, affecting bridges and signals.
Earth radius values you can choose
Most calculators use a mean Earth radius near 3,958.8 miles (6,371 km). A polar radius is smaller, about 3,949.9 miles, while an equatorial radius is larger, about 3,963.2 miles. Those differences change drop slightly, but not the overall trend. For quick checks, rounding R to 4,000 miles gives drop ≈ 0.66 feet × D², close to exact. That stays within one percent for planning.
Drop grows with distance squared
If you travel a distance D along the surface, the sagitta drop is h = R(1 − cos(D/R)). For short distances, the approximation h ≈ D²/(2R) is extremely accurate. Doubling the distance makes the drop about four times larger.
Rule of thumb in feet and inches
In U.S. customary units, a popular shortcut is about 8 inches of drop per mile squared. That means 1 mile ≈ 8 inches, 2 miles ≈ 32 inches, 5 miles ≈ 200 inches (about 16.7 feet), and 10 miles ≈ 800 inches (about 66.7 feet). In feet, the same shortcut is about 0.667 × D² feet.
Arc length versus line of sight
The calculator reports geometric drop from the starting tangent. A line of sight between two elevated points is different, because both endpoints can be above the surface. Horizon distance also depends on eye height; a 6‑foot observer sees the horizon around 3 miles away in still air. If both points are level, the greatest clearance issue is near the midpoint.
Refraction can hide real curvature
Real observations often include atmospheric refraction. Standard refraction bends light downward, effectively increasing the apparent radius and reducing the visible drop. Under temperature inversions, refraction can vary strongly, sometimes creating mirages that exaggerate or hide curvature effects.
Using the results for planning
For practical planning, enter your distance and pick units that match your map or measurement. Use the drop to estimate required clearance for a beam, antenna, or long straight alignment. For two endpoints, compare each endpoint’s height to the drop at mid‑distance.
Common unit pitfalls to avoid
Watch unit conversions closely: miles versus kilometers, feet versus meters, and radius modes. If you input distance in kilometers but read output in feet, errors can be huge often. Keep at least four significant digits when exporting results for engineering notes.
FAQs
1. What does “drop” mean in this calculator?
It is the sagitta: how far the Earth’s surface falls below a straight tangent line drawn at the start point over the chosen distance.
2. Is the “8 inches per mile squared” rule exact?
No. It is a close shortcut derived from the small‑angle approximation and a typical Earth radius. It is very accurate for a few miles and still useful for quick estimates at longer ranges.
3. Which Earth radius should I choose?
Use the mean radius for general work. Use equatorial or polar radii when you want a slightly tighter bound, or pick a custom radius if your standard specifies one.
4. Why do my real‑world observations look different?
Light bends in the atmosphere. Standard refraction reduces apparent curvature, while unusual temperature layers can increase bending or create mirages, changing what you see near the horizon.
5. Does height above ground change the curvature drop?
The geometric drop is based on distance and radius. Height matters for visibility and horizon distance, because elevated observers can see farther and clear more curvature.
6. Can I use kilometers and meters safely?
Yes, as long as units stay consistent. Choose kilometers for distance and a kilometer-based radius, then read outputs in meters or centimeters. Mixing mile inputs with metric outputs is the most common error.
7. What output is best for engineering notes?
Use the exact drop and the approximation together. The exact value is rigorous, while the approximation is handy for checking calculations and explaining results in reports.