Earth Curve Calculator - Calculator Academy

Calculate Earth curvature drop, horizon distance, and obscured object height from distance, eye level, radius, and refraction settings.

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Earth Curve Formula

The following short-distance and small-height approximations describe curvature drop and horizon distance. The calculator uses stable exact spherical geometry with the selected effective radius.

Drop = d² / (2R) Horizon = sqrt(2R \times h)

Where Drop is the vertical drop in meters
d is the distance in meters
R is Earth's radius (approximately 6,371,000 m)
Horizon is the distance to the horizon in meters
h is the observer's height in meters

For short arcs, drop ≈ d²/(2R). The exact drop below the starting tangent is R(1 − cos(d/R)); exact line-of-sight horizon distance is √(h(2R+h)). Distance d is a surface arc, and R is the selected effective radius.

What is an Earth Curve Calculator?

Definition:

An Earth Curve Calculator estimates the curvature of the Earth by determining the vertical drop over a given distance or by calculating the distance to the horizon from an observer’s height.

How to Calculate Earth Curve Measurements?

Example Problem:

The following example outlines the steps and information needed to calculate Earth's curve measurements.

First, consider a distance of 1000 m.

Next, calculate the drop using the formula above:

Drop = d^2 / (2R)

Drop = 1000^2 / (2 * 6371000)

Drop ≈ 0.0785 m

Then, consider an observer height of 2 m.

Now, calculate the horizon distance using the corresponding formula:

Horizon = sqrt(2R * h)

Horizon = sqrt(2 * 6371000 * 2)

Horizon ≈ 5048.17 m

FAQ

What does the Earth Curve Calculator compute?

It calculates the vertical drop due to Earth’s curvature based on distance and determines the horizon distance based on an observer’s height.

What assumptions are made in these calculations?

The default is a spherical Earth of radius 6,371,000 m with no refraction. You can change the radius or select a simplified effective-radius refraction assumption. Actual atmospheric variation and terrain are not modeled.

How should I use the calculator?

Choose Curvature drop, Horizon distance or Obscured object height. Enter the required surface distance or observer height, choose the result unit, and select Calculate. Model settings are optional; an object height enables visible/hidden portion results.