Why standing up lets you see farther.

Distance to the horizon uses the flat-horizon approximation d = √(2Rh), valid because eye height h is tiny compared to Earth's radius R (6,371 km) across almost this whole range. The d shown is the straight-line distance from your eye to the horizon point. The exact form of that line is d = √(2Rh + h²); dropping the h² term changes the answer by well under 1% at any human or airliner height, though by the top of this slider it matters more — at 500 km the figure here reads about 2% low, and the distance measured along the ground (2,445 km) is a further 3% shorter again than the sight line. This also ignores atmospheric refraction, which bends light slightly around the curve and extends the real visible horizon by roughly 7% beyond the geometric value calculated here. The hero diagram is a schematic, not to true scale — Earth's actual curvature over these heights is far too gentle to draw a visible arc at 1:1. See Wikipedia: Horizon — Distance to the horizon.