How much does the Earth curve per mile?

Approximately 8 inches (20 cm) for the first mile, but the drop grows quadratically — at 2 miles it is about 2.7 feet, and at 10 miles it is about 66.7 feet.

What is the difference between drop and bulge?

Drop is measured from a horizontal tangent line at the observer; bulge is the maximum height of the curved surface between two endpoints at the same elevation.

Does atmospheric refraction affect curvature visibility?

Yes. Standard atmospheric refraction effectively increases the Earth's radius by about 7%, bending light to follow the curvature slightly, which extends visibility.

Is the Earth a perfect sphere?

No. The Earth is an oblate spheroid, slightly flattened at the poles. The equatorial radius is about 21 km larger than the polar radius. For most practical calculations, a sphere of R = 6371 km is sufficiently accurate.

Can I see the curvature with my eyes?

From sea level, curvature is difficult to perceive directly. From high altitudes (10+ km) or in long-exposure photographs across large bodies of water, curvature becomes measurable.

How do surveyors account for curvature?

Surveyors apply curvature and refraction corrections to leveling observations. The combined correction is approximately 0.0675 × d² meters where d is in kilometers.

Earth Curvature Calculator

Calculate the curvature drop, bulge, and hidden height of objects over distance on a spherical Earth with optional refraction correction.

Distance

Distance Unit

Observer Height (m)

Target Height (m)

Refraction Coefficient

Curvature Drop

7.848 m

Vertical drop below a horizontal tangent at the observation point

Curvature Drop (ft)

25.75 ft

Same drop expressed in feet

Midpoint Bulge

1.962 m

Maximum bulge height of Earth's surface at the halfway point

Hidden Height

7.85 m

Height of distant object hidden behind the curvature

Target Visible?

No ✗

Target is completely hidden behind curvature

Observer Horizon

0.00 km

Distance to the geometric horizon from the observer

Curvature Drop vs Distance

1 km

0.08 m

2 km

0.31 m

5 km

1.96 m

10 km

7.85 m

20 km

31.39 m

50 km

196.20 m

100 km

784.81 m

Curvature Reference Table

Distance (km)	Drop (m)	Drop (ft)	Bulge (m)
1	0.08	0.26	0.02
2	0.31	1.03	0.08
5	1.96	6.44	0.49
10	7.85	25.75	1.96
20	31.39	102.99	7.85
50	196.20	643.71	49.05
100	784.81	2,574.82	196.20
200	3,139.22	10,299.29	784.85
500	19,620.15	64,370.59	4,906.93
1000	78,480.62	257,482.34	19,650.46

Planning notes, formulas, and examples

About the Earth Curvature Calculator

The earth curvature calculator computes how much the surface curves away from a horizontal reference line over a given distance. On a spherical Earth with radius 6,371 km, the surface drops by approximately 7.85 cm per kilometer at close range, following a quadratic relationship that grows rapidly with distance.

This calculator is indispensable for surveyors, civil engineers, and telecommunications professionals who must account for curvature effects in leveling, bridge construction, long-range photography, and line-of-sight radio links. Over a distance of just 10 km, the curvature drop exceeds 7.8 meters — enough to completely hide a two-story building from a ground-level observer.

The tool also calculates the midpoint bulge (the maximum height of the Earth's surface between two points), hidden height of distant objects, and whether a target of known height is visible from the observer's position. An optional atmospheric refraction coefficient allows more realistic modeling of optical line-of-sight under various atmospheric conditions.

When This Page Helps

Understanding Earth's curvature is crucial for long-distance engineering projects, surveying, telecommunications link budgets, and even settling common misconceptions about visibility. Whether you're designing a bridge, setting up a microwave relay, or calculating whether a distant cityscape should be visible across a lake, it lets you compare geometric curvature and atmospheric-refraction effects in one step.

How to Use the Inputs

Enter the horizontal distance you want to analyze
Select the distance unit (meters, kilometers, miles, or feet)
Optionally enter your observer height above the surface
Optionally enter the target object height to check visibility
Adjust the refraction coefficient for atmospheric effects (1 = no refraction, 1.07 = standard optical)
Review curvature drop, bulge, hidden height, and visibility results
Check the reference table for curvature values at standard distances

Formula used

Curvature drop: h = d² / (2·R) where d is horizontal distance and R is Earth's radius (6,371 km). Midpoint bulge: b = R − √(R² − (d/2)²). Hidden height: h_hidden = (d − d_horizon)² / (2R) where d_horizon = √(2·R·h_observer). Refraction uses effective radius R_eff = k·R.

Example Calculation

Result: 7.85 m curvature drop at 10 km

At 10 km, the drop is (10000)² / (2 × 6,371,000) = 7.85 m. A person standing at ground level would not see any object shorter than 7.85 m at that distance.

Tips & Best Practices

The approximate formula "8 inches per mile squared" is a popular shorthand (drop in inches ≈ 8 × miles²)
For surveying, use the combined curvature-refraction coefficient of 0.0675·d² (km) for the net correction
At distances under 1 km, curvature effects are negligible for most practical purposes
Use refraction coefficient 1.07 for standard visual and 1.33 for radio/radar calculations
Large bodies of calm water are ideal for observing curvature effects in photography

When To Use This Calculator

Calculate the curvature drop, bulge, and hidden height of objects over distance on a spherical Earth with optional refraction correction. Use it when you need a repeatable calculation in the physics / general category and want the setup, result, and supporting values kept together. This is especially helpful when small input changes, unit choices, or rounding decisions can change the final number.

How To Check The Result

Start by confirming that the inputs match the formula shown on the page. Then compare the main output with the worked example and any secondary values shown by the calculator. If the result will be used in another calculation, keep extra precision until the final step and record the assumptions beside the number.

Practical Notes

Treat the result as a calculation aid rather than a substitute for context. For schoolwork, include the formula and substitution steps. For planning, technical, financial, or health-related decisions, verify important numbers against primary records, current rules, or a qualified professional before acting on them.

Sources & Methodology

Last updated: January 15, 2025

Frequently Asked Questions

Approximately 8 inches (20 cm) for the first mile, but the drop grows quadratically — at 2 miles it is about 2.7 feet, and at 10 miles it is about 66.7 feet.