Earth Curvature Calculator

Earth curvature affects how distant objects appear relative to a local horizontal line. This calculator models Earth as a sphere with a mean radius of 6,371 kilometers. From an observer's height it finds the straight-line geometric distance to the horizon. From a surface range it also finds how far the spherical surface falls below the tangent plane at the starting point. Horizon distance follows from a right triangle whose hypotenuse runs from Earth's center to the observer. The exact geometric expression is the square root of 2Rh plus h squared, where R is Earth radius and h is observer height. Near the surface, the h-squared term is tiny, but retaining it keeps the equation valid for taller observations. Increasing height does not increase horizon distance linearly: horizon range grows approximately with the square root of height. Curvature drop is calculated with exact circle geometry rather than the common eight-inches-per-mile approximation. For a horizontal chord distance d, the drop is R minus the square root of R squared minus d squared. The calculator interprets the entered range as the horizontal distance for this model. At short distances, drop is approximately d squared divided by twice Earth's radius, so doubling distance produces about four times the drop. Hidden height here is a simple geometric comparison between curvature drop and observer elevation. It subtracts observer height from the calculated drop and never reports a negative value. It is useful for a first estimate, but it is not a complete line-of-sight solution for an elevated target. A rigorous visibility calculation should include both observer and target heights, terrain, tides, atmospheric conditions, and the exact geodetic path. Atmospheric refraction commonly bends light downward and extends the apparent horizon beyond the purely geometric value. Refraction varies with pressure, temperature, humidity, and vertical gradients, so this calculator deliberately reports unrefracted geometry. Surveying, radio planning, navigation, and photography may use an effective Earth radius such as seven-sixths of the actual radius under standard assumptions. Treat that as a separate model rather than a guaranteed correction.