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Spherical Body Geometric Horizon Distance

Calculate the straight-line tangent distance to an ideal spherical horizon from height and radius.

  • Formula & worked example
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Calculator inputs

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How to use this calculator

  1. Enter the known values in the units shown. Results update as you type.
  2. Where results are editable, change one to solve backwards. Lock a value to hold it fixed.
  3. Use the worked example to check the method. Reset restores the starting fields.

Private by default

Inputs and results stay in this browser tab. Bookify does not upload or store the values you enter.

Formula and method

For a sphere of radius 4 km and an observer 1 km above it, the straight-line distance to the tangent horizon is 3 km.

Straight-line horizon distance = √(height × (height + 2 × radius))

Worked example

Enter these known values and leave the other values blank.

Reference sphere
Enter a custom radius
Custom sphere radius
4 km
Height above the ideal spherical surface
1000 m
Straight-line distance to the horizon
3 km

Assumptions and limitations

  • Height, radius and distance must be positive. The calculation comes from a right triangle between the center, observer and tangent point.
  • The answer is line-of-sight distance, not the arc length along the surface. Atmospheric refraction, terrain, oblateness and obstructions are omitted.
  • Named bodies use fixed approximate radii, including Earth at 6,371 km. Gas giants and the Sun are reference spheres, not solid observation surfaces. Use Custom to supply another radius.

Common questions

Does this predict the visible horizon in all weather?

No. Refraction can bend the light path, and terrain or haze can limit visibility before the geometric horizon.

Can I find height from a known distance?

Yes, with a fixed radius the model solves h = √(d² + R²) − R. The distance must use the same straight-line definition.

References

The calculation equations, inverse formulas, units, and input rules were imported from this source. Bookify provides the interface and equation solver.

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