PhysicsCalcs
Optics

Distance to Horizon Calculator

Calculate how far you can see from any height — with curvature drop, refraction correction, and planetary horizon modes.

Interactive calculator

Distance to Horizon Calculator

Calculate how far you can see based on height, with modes for curvature drop, dip angle, refraction correction, two-object visibility, and planetary horizons.

Try an example

Your result will appear here.

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

Quick Guide

  • Choose: horizon distance, two-object visibility, curvature drop, dip angle, refraction-adjusted, or planetary.
  • Use presets for common scenarios (standing, lighthouse, aircraft, Everest).
  • Click Calculate to see detailed results.

Table of Contents

Key Takeaways

  • Horizon distance ≈ √(2Rh) where R is Earth's radius and h is observer height.
  • At eye level (1.7 m), the horizon is about 4.7 km (2.9 miles) away.
  • Atmospheric refraction extends the visible horizon by ~8% under standard conditions.
  • Two elevated objects can see each other beyond the sum of their individual horizons.
  • On the Moon or Mars, horizons are closer due to smaller planetary radii.

How Far Can You See?

The distance to the horizon depends on your height above the surface and the planet's radius. On a perfectly clear day with no obstructions, the horizon is the farthest point where the Earth's surface is tangent to your line of sight. For a person at sea level (1.7 m), this is about 4.7 km.

Horizon Formula

d=(R+h)2−R2≈2Rhd = \sqrt{(R+h)^2 - R^2} \approx \sqrt{2Rh}

Where R is the planet's radius (6,371 km for Earth), h is observer height, and d is the line-of-sight distance to the horizon. The approximation works well when h is much smaller than R.

Atmospheric Refraction

dref=2Reff⋅h,Reff=R1−kd_{\text{ref}} = \sqrt{2R_{\text{eff}} \cdot h}, \quad R_{\text{eff}} = \frac{R}{1-k}

The atmosphere bends light downward, letting you see slightly farther than geometry alone predicts. The standard refraction coefficient k = 0.13 extends the horizon by about 8%. Surveyors use k = 0.17 as a standard correction factor.

How to Use

  1. Select a mode from the dropdown.
  2. Enter height (and other values as needed).
  3. Click Calculate. Results show exact and approximate values.

Examples

Person at beach (h = 1.7 m)

d = √(2 × 6,371,000 × 1.7) ≈ 4,654 m ≈ 4.7 km

Aircraft at 10 km

d = √(2 × 6,371,000 × 10,000) ≈ 357 km (222 miles)

FAQ

How far can a person see?›

A person standing at 1.7 m (5'7") above sea level can see approximately 4.7 km (2.9 miles) to the geometric horizon. With standard atmospheric refraction (k = 0.13), this extends to about 5.0 km.

What is the curvature drop?›

The curvature drop is how far the Earth's surface falls below a level line over a given horizontal distance. The approximation is drop ≈ d²/(2R), which gives about 8 inches per mile squared — a useful surveying rule of thumb.

What is the refraction coefficient k?›

The refraction coefficient k accounts for how atmospheric bending curves light. Standard k = 0.13 means light bends about 13% of Earth's curvature, extending the visible horizon. This varies with temperature, pressure, and humidity.

How does height affect the horizon?›

Horizon distance grows with the square root of height. Doubling your height only increases the horizon by about 41%. From 10 km altitude (aircraft), you can see about 357 km.

Is the formula different for other planets?›

The same geometry applies. Just substitute the planet's radius: Moon (1,737 km), Mars (3,390 km), etc. Smaller planets have closer horizons. Atmospheric refraction only applies to bodies with atmospheres.

Sources

  1. Wikipedia – Horizon
  2. Andrew T. Young – Dip of the Horizon
  3. NOAA – Geodetic Computations
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