Viewshed Radius Calculator
An observer's viewshed on smooth terrain is limited by the curve of the Earth. The higher you stand, the farther the horizon. This calculator returns the geometric horizon distance for a given eye height, then applies an atmospheric refraction factor to give the effective visible horizon. Use it to size a line-of-sight radius for an antenna, lookout tower, drone or camera before adding terrain detail. The radius is reported in kilometres and miles.
Horizon distance formula
R_eff = Earth radius * refraction factor
geometric horizon = sqrt(2 * R * h + h^2)
effective horizon = sqrt(2 * R_eff * h + h^2)
viewshed area = pi * (effective horizon)^2
The horizon distance comes from the right triangle between the Earth's centre, the observer's eye and the tangent point. Refraction is handled by inflating the Earth radius, the standard surveying approximation. The viewshed area treats the horizon as a circular radius on flat ground.
Worked example
Eye height 10 m, R = 6,371.0088 km (6,371,008.8 m), refraction 1.17. Geometric horizon = sqrt(2 times 6,371,008.8 times 10 + 100) = 11,287 m = 11.29 km. Effective horizon with R times 1.17 = 12.21 km, about 7.59 miles. The viewshed area is pi times 12.21^2 = 468.3 km2.
Viewshed radius: frequently asked questions
What is a viewshed radius?
The viewshed radius here is the distance to the geometric horizon: the farthest point on a smooth Earth that an observer at a given height can see before the curvature of the planet hides the ground. It sets the outer limit of a line-of-sight viewshed on flat terrain.
What formula is used for horizon distance?
The geometric horizon distance d = sqrt(2 * R * h + h^2), where R is the Earth's radius and h is the eye height. Because h is tiny compared with R, this is very close to d = sqrt(2 * R * h). Atmospheric refraction extends the visible horizon, modelled by multiplying R by a refraction factor (about 7/6).
Why include atmospheric refraction?
Light bends slightly downward through the atmosphere, letting you see a little beyond the geometric horizon. Standard surveying practice uses an effective Earth radius about 1.15 to 1.17 times the true radius (the 7/6 rule). This calculator lets you set the refraction factor, defaulting to 1.17.
Does terrain affect the real viewshed?
Yes. This calculator gives the smooth-Earth horizon. Hills, buildings and target height all change a real viewshed. To include a target's height, compute each horizon distance separately and add them: total range = horizon(observer) + horizon(target).
Sources and references
- National Oceanic and Atmospheric Administration: atmospheric refraction reference.
- U.S. Geological Survey: elevation and terrain data.
- Formula: spherical-Earth horizon geometry with the standard 7/6 refraction approximation.
Reviewed by the CalculatorHub team, edited by James Graham, 19 June 2026. See our methodology.