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Horizon Distances

How far away can an object be before it's blocked by the horizon?

Posted: Feb 15th, 2023

How far away can an object be before it dips below the horizon and is no longer visible?

For simplicity, treat the earth like a smooth perfect sphere, and ignore the atmosphere and any other obstacles.


Maximum distance at which object is visible: Arc distance along ground from feet to base: meters. ( miles)
Straight-line distance from eyes to tip of object: meters. ( miles)

Table of Examples

Assuming that your eyes are 1.8 meters above the ground, (and the earth is perfectly spherical etc. etc.) from how far away can you see the tops of various objects?

Object Object Height (m) Visable Distance (m) Vis. Dist. (miles)
The Ground Itself 0 4,800 3.0
Garden Gnome 0.5 7,300 4.5
Another Person 1.8 9600 6.0
A Male Giraffe 5.2 13,000 8.0
An Aspen Tree 16 19,000 12
Statue of Liberty (with base) 93 39,000 24
Great Pyramid of Giza 140 47,000 29
Statue of Unity (with base) 240 60,000 37
Burj Khalifa 830 110,000 67
Mount Everest 8800 340,000 210
Severe Thunderstorm 18000 480,000 300

Arc distance along the ground is same as straight-line distance, up to the precision I used for these numbers. Once you get up to the Mount Everest row, the difference is on the order of one mile.


Straight-line Distance

Assume the earth is a perfect sphere with radius $R_⊕$.

If your eyes are at height $h$ above the surface of the sphere, then the distance to the horizon is:

\[d = \sqrt{ {(R_⊕+H)}^2 - R_⊕^2}\]

A diagram showing how a right angle is formed between an Earth radius and the horizon sight line.

Now suppose there’s an object of height $H$ which is so far away that you can just barely see the top of it over the horizon.

The distance from the horizon to the top of that object is

\[D = \sqrt{ {(R_⊕+H)}^2 - R_⊕^2 }\]

And the distance from your eyes to the top of that object is

\[distance = d + D = \sqrt{ {(R_⊕+h)}^2 - R_⊕^2} + \sqrt{ {(R_⊕+H)}^2 - R_⊕^2 }\]

A diagram extending the horizon line to a faraway object.

Distance Along the Ground

Unless you’re looking at something really tall, the arc-length distance along the ground will be roughly the same as the straight-line distance.

But it you want to know the exact distance along the ground, the formula is

\[arclength = (\theta_1 + \theta_2) \cdot R_⊕\]

Where the angles are

\[\theta_1 = \arccos \left(\frac{R_⊕}{R_⊕+h}\right)\] \[\theta_2 = \arccos \left(\frac{R_⊕}{R_⊕+H}\right)\]
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