did you know you can calculate how far away the horizon is?

stand on a beach with your eyes about 1.7 m above the water, and the horizon is roughly 5 km away. that’s all. the sea looks endless, but the curve cuts you off quickly.

climb a 100 m cliff and the horizon retreats to about 36 km. height buys distance, fast.

draw a line from the centre of the earth to your eyes, and another from your eyes to the horizon. that second line just grazes the surface — a tangent.

the tangent, the radius to the grazing point, and the line to your eyes form a right triangle. pythagoras does the rest.

distance ≈ √(2 × R × h) — about 3.57 × √h km, with h in metres

R is earth’s radius, about 6,371 km. ignores atmospheric refraction, which bends light slightly and stretches the horizon a little further.

by pythagoras: (R + h)² = R² + d². expand and you get d² = 2Rh + h².

because h is tiny next to R, the h² term barely matters, so d ≈ √(2Rh). plug in earth’s radius and the 3.57√h rule falls out.

a beautiful habit in physics: keep the big terms, drop the small ones, and check the error is harmless.

(R + h)² = R² + d² → d ≈ √(2Rh)

• this is why ships’ masts appear before their hulls — and one of the oldest clues that the earth is round.
• refraction typically adds about 8% to the distance, which is why lighthouses are visible a little further than the formula says.
• on the moon the horizon is closer; on a bigger planet it would be further. the formula travels well.