PhysicsCalcs
Optics

Earth Curvature Calculator

Calculate curvature drop, hidden height, bulge, line-of-sight, and refraction-adjusted values.

Interactive calculator

Earth Curvature Calculator

Calculate curvature drop, hidden height, bulge, line-of-sight distance, and refraction-adjusted values for Earth's curvature.

Try an example

Your result will appear here.

Choose a calculation mode, fill in the known values, and click Calculate.

Quick Guide

  • Choose: curvature drop, hidden height, horizon, bulge, line-of-sight, or refraction-adjusted.
  • Enter distances in metres, km, or miles.
  • Click Calculate for detailed results.

Table of Contents

Key Takeaways

  • Curvature drop ≈ d²/(2R) — the surface falls away quadratically with distance.
  • The "8 inches per mile squared" rule is a useful approximation for short distances.
  • The sagitta (bulge) at the midpoint of a chord is approximately d²/(8R).
  • Hidden height tells you how much of a distant object is blocked by the curvature.
  • Atmospheric refraction reduces apparent curvature by about 13% under standard conditions.

Earth's Curvature

Earth is approximately spherical with a mean radius of 6,371 km. This curvature is imperceptible at small scales but has significant effects at distances of a few kilometres and beyond — hiding distant objects, requiring surveying corrections, and limiting radio line-of-sight.

Curvature Formulas

Δh=R−R2−d2≈d22R\Delta h = R - \sqrt{R^2 - d^2} \approx \frac{d^2}{2R}
sagitta=R−R2−(d/2)2≈d28R\text{sagitta} = R - \sqrt{R^2 - (d/2)^2} \approx \frac{d^2}{8R}

Where R = 6,371 km is Earth's mean radius, d is the horizontal distance along the surface, and Δh is the vertical drop below a level tangent plane.

8 Inches per Mile Squared

The classic surveying rule states the curvature drop is approximately 8 inches per mile squared. At 1 mile: 8", at 2 miles: 32", at 3 miles: 72". This approximation is accurate to within 1% for distances up to about 100 miles.

How to Use

  1. Select a calculation mode.
  2. Enter distances and/or heights.
  3. Click Calculate for exact and approximate results.

Examples

Drop over 10 km

Δh ≈ (10,000)² / (2 × 6,371,000) ≈ 7.85 m

Drop over 1 mile

Δh ≈ (1,609.34)² / (2 × 6,371,000) ≈ 0.203 m ≈ 8 inches

FAQ

How much does the Earth curve per mile?›

Using the approximation drop ≈ d²/(2R), the Earth drops about 8 inches (20 cm) over 1 mile. Over 2 miles it's about 32 inches (4× as much, since it scales with d²). This is the famous '8 inches per mile squared' rule.

What is the sagitta?›

The sagitta (or bulge) is the maximum rise of a circular arc above the chord connecting two points. For Earth's surface, it's the height of the midpoint above a straight tunnel between two surface points: s ≈ d²/(8R).

Can I see a building 50 km away?›

It depends on your height and the building's height. Use the 'Hidden Height' mode — enter the distance and your eye height. The calculator tells you how much of the building is hidden. A tall building may still be partially visible even at 50 km.

Does refraction change the curvature?›

Refraction doesn't change physical curvature, but it changes apparent curvature. Standard atmospheric refraction (k = 0.13) makes the Earth appear less curved, reducing the apparent drop by about 13%.

Sources

  1. Wikipedia – Earth's Curvature
  2. Surveying Mathematics – Curvature and Refraction
Manish Kumar

Author & technical reviewer

Manish Kumar

PhysicsCalcs tools are reviewed with an educational focus: clear formulas, transparent assumptions, and practical context for students and science learners.

Learn more about Manish