How the Distance to Horizon Is Calculated
The horizon is the farthest point you can see across a curved Earth before the surface drops below your line of sight. Because the Earth is (approximately) a sphere, the sightline from your eye just grazes the surface tangentially — a geometry that reduces to a simple right triangle. This calculator uses that geometry, plus a standard correction for how the atmosphere bends light, to compute how far you (and an optional distant object) can see.
Deriving the formula from Earth's curvature
Picture a right triangle formed by Earth's center, your eye at height h above the surface, and the point on the horizon where your line of sight touches the surface tangentially. The hypotenuse is R + h (Earth's radius plus your height), one leg is R (Earth's radius, from the center to the tangent point), and the other leg is the horizon distance d — because a tangent line always meets a radius at a right angle. By the Pythagorean theorem, d² + R² = (R + h)², which expands to d² = 2Rh + h², so d = √(2Rh + h²). Since h is almost always tiny compared to R (a few meters versus Earth's 6,371 km radius), the h² term is negligible and d ≈ √(2Rh) is accurate to a fraction of a percent for any height a person could stand at.
Why atmospheric refraction matters
Light does not travel in a perfectly straight line through Earth's atmosphere — it bends very slightly toward the surface because air density (and its refractive index) decreases with altitude. Under standard atmospheric conditions, this bending curves the light ray enough that it behaves as if Earth's radius were about 7/6 (roughly 1.14) times larger than it actually is. Plugging that effective radius into the same formula pushes the visible horizon about 7-8% farther out, which is why this calculator lets you toggle refraction on or off. Refraction is a statistical average, though — temperature inversions, fog, and unusual weather can shrink or exaggerate it in the real world.
Real-world uses and limits
- Mariners and pilots use it to estimate how far away a ship, coastline, or another aircraft first becomes visible.
- Lighthouse and tower designers add the target's own height to get a light's nominal geographic range, exactly what the "Maximum Line-of-Sight Range" result above computes.
- Radio and microwave line-of-sight links use a different constant (the "4/3 rule") because radio waves refract differently through the atmosphere than visible light.
- The formula assumes a smooth, spherical Earth and a perfectly clear line of sight — it ignores terrain, buildings, haze, and local weather, so real-world visibility is often shorter than the calculated geometric limit.