Earth Curvature Calculator
Earth curvature calculator — compute curvature drop (h = d²/2R), test the “8 inches per mile²” rule, bulge height, and the part of objects hidden behind curvature. With atmospheric refraction option.
Scope of this tool: this tool (#1595) computes curvature drop & bulge at flat distance d (how far the surface “drops” below the horizontal line). For the reverse question — from observer height h, how far away the horizon is (d = √(2Rh)) — use the sibling tool:
Distance to Horizon Calculator →
Example: 1 mile distance → drop 8.00 inches (20.33 cm; 0.2033 m).
same unit as above
One unit is used for both distances. Example: 5 + 5 miles → middle bulge 5.08 m (16.67 feet).
Skema geometri
Drop reference table (computed live)
Kilometers
| Distance | Drop | Drop | Drop |
|---|
Miles
| Distance | Drop | Drop | Drop |
|---|
R = 6,371,000 m, ideal sphere model without refraction.
Methodology (honest)
drop h = d² / (2R), R = 6.371.000 m
inverse: d = √(2R · h)
tonjolan: H = d1 × d2 / (2R) (simetris: H = drop/4)
tersembunyi: (d − dh)² / (2Reff), dh = √(2Reffh)
refraksi: Reff = R / (1 − k), k = 0 / 0,13 / 0,17
- Ideal sphere model. Earth is actually slightly oblate (equator ±6,378 km, poles ±6,357 km) and has relief; for practical distances the difference is below field-measurement uncertainty.
- The “8 inches per mile²” rule is an approximation — always ≈0.031% below the exact d²/(2R) value. Shown side by side for transparency.
- Refraction reduces the apparent effect (horizon farther, less hidden); choose the k assumption explicitly in the Hidden Object tab.
- Not for professional surveying — no geoid, topography, or tide corrections.
Frequently asked questions (FAQ)
What is Earth curvature "drop"?
Drop is how far the Earth’s surface "falls" below a straight horizontal line drawn from the observer, measured at flat distance d. The formula is h = d²/(2R) with R = 6,371,000 m (ideal sphere model). Example: at 1 mile (1.609 km), drop = 0.2033 m ≈ 20.33 cm ≈ 8.00 inches. At 1 km, drop = 7.85 cm; at 10 km, drop = 7.85 m.
Is it true Earth curves "8 inches per square mile"?
That is a rule of thumb (an approximation), not a law of physics. The exact value of d²/(2R) at 1 mile² is 8.0025 inches — so the "8 inch" rule is always ≈0.031% below the exact formula, at any distance. So small a difference can be ignored for rough estimates (at 10 miles: exact 800.25 inches vs approximate 800 inches, a 0.25-inch difference), but for precision work always use the exact formula. This calculator shows both side by side so you see the difference honestly.
How does this calculator differ from a distance-to-horizon calculator?
This calculator (#1595) computes curvature drop/bulge at FLAT DISTANCE d: how far the surface drops (h = d²/2R), the bulge height midway between two points, and how much of an object is hidden. The Distance to Horizon calculator (#1590, https://nrtechstudio.com/tools/distance-to-horizon/) answers the reverse question: from OBSERVER HEIGHT h, how far away the horizon is (d = √(2Rh)), including inverse and two-point modes. The two complement and link to each other.
What is bulge height?
Bulge is how high the Earth’s surface "bulges" at the midpoint between two points at distances d1 and d2 from the measuring point. The formula is H = d1×d2/(2R). For the symmetric case (d1 = d2), bulge = drop/4. Example: two points 10 miles apart in total (5 miles + 5 miles) have a middle bulge of 5.08 m (16.67 feet). This mode is useful for planning radio/microwave line-of-sight paths and long-distance surveys.
How do you calculate the part of an object hidden by curvature?
From eye height h, first compute your horizon distance: √(2Rh). Eyes at 1.7 m see the horizon at 4.65 km (no refraction) or 4.99 km (standard refraction). Objects FARTHER than that start to hide: the lower (d−dh)²/(2R) is not visible. Example: a ship 10 km from a 1.7 m observer hides 2.24 m of its lower part (1.71 m with standard refraction) — which is why a ship’s hull "sinks" first.
Does atmospheric refraction affect the results?
Yes. Air bends light downward so Earth looks slightly larger than it is. The model: effective radius Reff = R/(1−k). The standard value k = 0.13 (normal conditions) extends the horizon and REDUCES the hidden height; k = 0.17 for strong refraction. This calculator offers three choices (none, standard, strong) to keep your assumptions transparent, not hidden behind the numbers.
Is Earth assumed a perfect sphere here?
Yes — this calculator uses an ideal sphere model with radius 6,371,000 m, stated explicitly. Earth is actually slightly oblate: equatorial radius ±6,378 km, polar radius ±6,357 km, plus local topography. For practical distances under hundreds of kilometers, the difference from this model is far smaller than field-measurement uncertainty such as tide heights and daily refraction variation.
Up to what distance is this formula valid?
The parabola formula h = d²/(2R) is accurate as long as the distance is much smaller than Earth’s radius — practically up to hundreds of kilometers (at 100 km, drop = 784.81 m). For thousands of kilometers, the curve can no longer be approximated by a parabola and you need full spherical geometry. This calculator rejects distance inputs above 20,000 km to avoid misleading numbers.
Which units are supported?
Distance: miles, kilometers, and meters. Drop/height results: inches, centimeters, meters, and feet — all four always shown together so no manual conversion is needed. Inverse mode accepts a target drop in inches, centimeters, meters, or feet, then computes the distance that produces it.
Why do ships at sea look like their hulls "sink" first?
Because the ship’s lower part enters the hidden zone first. The hidden zone grows quadratically with distance ((d−dh)²/(2R)), so the hull (near the water surface) disappears first, the deck follows, and the mast or tallest structure stays above the horizon longest. This phenomenon is observed daily by sailors and is a curvature demonstration anyone can verify with binoculars.
Can these results be used for professional surveying?
Not directly. This is an educational model: ideal sphere, no topography, no tides, and highly simplified refraction. Professional surveying uses geoid corrections, field elevation data, and actual refraction measurements. Use this calculator to understand orders of magnitude and rough planning, not as a substitute for survey instruments.
Where can I compute horizon distance from my elevation?
In this tool’s sibling calculator: the Distance to Horizon Calculator at https://nrtechstudio.com/tools/distance-to-horizon/ — enter eye or tower height, get the horizon distance, complete with refraction options, inverse mode (height from observed distance), and two-point mode for lighthouses or transmitter towers.
Free tool by NRTechStudio