Manually Calculating the Amount Hidden by Curvature

In order to calculate the amount that should be hidden we may use the Theory of Pythagoras.

Diagram & Transformations Credit: dizzib | Theory Credit: Pythagoras

Important Equations

To find the distance from the observer's eye to the horizon (d1) we first perform the following:

1. d1 = sqrt(h0^2 + 2 * R * h0)

Next, we place d1 in the following equation to find the amount hidden below the horizon of the earth (h1):

2. h1 = sqrt((d0 - d1)^2 + R^2) - R

Example 1: Sea Level

To calculate the amount hidden at sea level, over 6.23 miles, with an observer height of 32 inches, we convert to a like unit (e.g. km) and perform the following:

R = 6371 km
h0 = 0.0008128 km (32 inches)
d0 = 10.02621 km (6.23 miles)

1. d1 = sqrt(0.0008128^2 + 2×6371×0.0008128) = 3.21818
2. h1 = sqrt((10.02621 - 3.21818)^2 + 6371^2) - 6371 = 0.00363752

0.00363752 km converted to feet = 11.93412073491 feet hidden

Example 2: Lake Above Sea Level

To calculate the amount hidden across a lake with an altitude of 1368 meters above sea level, we make a slight adjustment to R and perform the following:

R = 6372.368 km (6371 km + 1368 m)
h0 = 0.0008128 km (32 inches)
d0 = 10.02621 km (6.23 miles)

1. d1 = sqrt(0.0008128^2 + 2×6372.368×0.0008128) = 3.21853
2. h1 = sqrt((10.02621 - 3.21853)^2 + 6372.368^2) - 6372.368 = 0.00363636

0.00363636 km converted to feet = 11.93031496063 feet hidden

Online Calculator

Based on the above, dizzib has provided an online calculator for computing the drop distance and the amount which should be hidden per observer height according to the curvature of the earth.