Why eye height changes how far you can see
How the square root relationship between eye height and horizon distance works, and how to combine two heights to spot ships and lights.
The square root relationship
Distance to the horizon does not grow in step with your height. Because the geometry involves the square root of eye height, gaining altitude gives diminishing returns: going from 9 feet to 36 feet, a fourfold increase, only doubles the range. That is why a crow's nest was such a prize on a sailing ship, and why even a modest climb up the mast noticeably extends what a lookout can see. The calculator makes this concrete, showing 5.91 km from 9 feet but only 13.94 km from 50 feet.
Adding two horizons to spot an object
The horizon is a shared boundary, so two observers or an observer and a tall object both see over the same curve of the sea. To find how far off a lighthouse first appears, work out the horizon distance for your own eye height and the horizon distance for the light's height, then add the two. A 50 foot light and a 9 foot eye height, for instance, become visible to each other at roughly the sum of their individual ranges. This is exactly how a light's charted geographic range is derived.
Refraction and why the view is fuzzy
The atmosphere is not uniform, and light passing through it bends toward the denser, cooler air near the surface. This refraction lets you see slightly beyond the true geometric horizon, which is why the tool uses 3.57 rather than the geometric 3.83. On days with strong temperature layering the effect grows or reverses, producing looming, sinking or a shimmering mirage at the horizon line, so treat any single number as a good average rather than a guarantee.
Putting the distance to practical use
Knowing your horizon range helps you judge whether a sail on the edge of sight is closing or holding station, and it sets a realistic limit on visual piloting before you rely on radar or charts. It also frames photography and stargazing over water, where the horizon marks the lowest a rising object can be seen. Because the model only needs one input, it is quick to rerun as you move from the cockpit to the coach roof and watch the horizon step outward.