To find the distance at 53°S, we multiply this equatorial distance by the cosine of the latitude.
- Globe-based result: 66.90 km
- Flat earth result: 67.01 km
- Absolute difference: 0.11 km (110 meters)
- Percentage difference: 0.16%
The use of the cosine function in our formula comes from observational evidence. When we measure the apparent distances between meridians of longitude at different latitudes, we notice they appear to get closer together as we move away from the equator, following a pattern that matches the cosine function.
This relationship can be verified through direct measurement. If we place markers at one-degree intervals along different latitude lines, we observe that the distances between these markers decrease in proportion to the cosine of the latitude angle. For example:
At the equator (0°): The full distance of 111.32 km = 111.32 × cos(0°)
At 30°N/S: About 96.4 km = 111.32 × cos(30°)
At 53°S: About 67.01 km = 111.32 × cos(53°)
This consistent pattern appears in surveying data across all latitudes. The cosine function naturally describes how the apparent spacing between meridians changes from an observer’s perspective on a flat plane. Just as the cosine of an angle represents the adjacent side of a right triangle divided by the hypotenuse, it captures how the observable distance between meridians relates to the maximum distance at the equator.
The beauty of this approach is that it emerges directly from measurement and observation, without requiring complex theoretical frameworks. The cosine relationship provides a simple, elegant way to calculate distances that matches what surveyors actually measure in the field.