Earth Curvature Calculator
Calculate the vertical drop due to Earth's curvature over any distance
Distance over which to calculate curvature drop
How to Use This Calculator
Enter the Distance
Input the horizontal distance in meters over which you want to calculate Earth's curvature. For example, 1000 m (1 km), 5000 m (5 km), or 10000 m (10 km).
Calculate
Click the "Calculate Curvature Drop" button to get the vertical drop (how much Earth curves downward) over that distance. The result is shown in meters, centimeters, and feet.
Interpret Results
The result shows how much the Earth's surface drops below a perfectly flat horizontal line over the specified distance. This is useful for understanding visibility, surveying, and navigation.
Formula
h = d² / (2R)
Where:
- h = Curvature drop (vertical drop due to Earth's curvature, in meters)
- d = Horizontal distance (in meters)
- R = Earth's radius ≈ 6,371,000 meters (6,371 km)
Note: This is an approximation valid for distances much smaller than Earth's radius. For very large distances, the exact formula accounting for the arc length should be used.
Example Calculation (1 km distance):
For a distance of 1,000 meters (1 km):
d = 1,000 m
R = 6,371,000 m
h = (1,000)² / (2 × 6,371,000)
h = 1,000,000 / 12,742,000
h = 0.0785 m
h ≈ 7.85 cm (about 3 inches)
Example Calculation (10 km distance):
For a distance of 10,000 meters (10 km):
d = 10,000 m
h = (10,000)² / (2 × 6,371,000)
h = 100,000,000 / 12,742,000
h = 7.85 m
Over 10 km, Earth curves about 7.85 meters.
About Earth Curvature Calculator
Earth's curvature refers to the fact that our planet is spherical, causing the surface to drop below a perfectly flat horizontal line as distance increases. This calculator determines how much vertical drop occurs over a given horizontal distance due to Earth's curvature. Understanding Earth's curvature is essential for navigation, surveying, construction of long structures, and understanding why objects disappear over the horizon.
When to Use This Calculator
- Surveying: Account for Earth's curvature when measuring long distances
- Construction: Plan long bridges, tunnels, and pipelines that must account for curvature
- Navigation: Understand how Earth's curvature affects visibility and radar range
- Astronomy: Calculate how curvature affects observations and horizon calculations
- Educational Purposes: Demonstrate that Earth is spherical and understand geometric principles
Why Use Our Calculator?
- ✅ Instant Results: Get accurate curvature calculations immediately
- ✅ Easy to Use: Just enter the distance to get the drop
- ✅ Multiple Units: Results displayed in meters, centimeters, and feet
- ✅ Accurate Formula: Uses the standard approximation for Earth's curvature
- ✅ Educational: Includes formula explanations and worked examples
- ✅ 100% Free: No registration or payment required
Common Applications
Long-Distance Construction: Engineers building long bridges, pipelines, or tunnels must account for Earth's curvature. For example, over 10 km, the curvature drop is about 7.85 meters, which can significantly affect construction tolerances and alignment.
Surveying and Mapping: Surveyors use curvature calculations to correct measurements over long distances. Without accounting for curvature, long-distance measurements would be inaccurate, affecting maps, property boundaries, and construction projects.
Understanding Visibility: Earth's curvature explains why objects disappear over the horizon. A ship 10 km away will be hidden by about 7.85 meters of curvature, which is why tall objects like lighthouses are needed for navigation.
Tips for Best Results
- Use distance in meters for the most accurate calculations
- Remember that curvature increases with the square of distance—doubling distance quadruples the drop
- For distances less than 100 km, the approximation formula is very accurate
- The formula assumes a perfect sphere; actual Earth has slight flattening at the poles
- Atmospheric refraction can affect apparent curvature, but this calculator gives the geometric value
- For very precise work over long distances, consider using the exact arc formula
Frequently Asked Questions
Why does curvature increase with the square of distance?
The curvature formula h = d²/(2R) shows that the drop is proportional to the square of distance. This is because as you move further along a curved surface, the deviation from a straight line accumulates quadratically. Doubling the distance means the curvature drop increases by a factor of four (2² = 4).
Is Earth a perfect sphere?
No, Earth is an oblate spheroid—slightly flattened at the poles and bulging at the equator. However, for most practical purposes, especially over moderate distances, treating Earth as a sphere with radius 6,371 km is sufficiently accurate. The difference between equatorial and polar radius is only about 21 km.
How does this relate to the "8 inches per mile" rule?
The common approximation "8 inches per mile squared" is derived from this formula. For 1 mile (1,609 m), the curvature is about (1,609)²/(2×6,371,000) ≈ 0.203 m ≈ 8 inches. Over 2 miles, it's about 32 inches (2² × 8), confirming the "per mile squared" relationship.
Does this account for atmospheric refraction?
No, this calculator gives the geometric curvature only. Atmospheric refraction can bend light slightly, making objects appear higher than they geometrically are. Under standard conditions, refraction can extend visibility by about 8%, but the geometric calculation remains the baseline for understanding Earth's shape.
Why is understanding Earth's curvature important?
Understanding Earth's curvature is crucial for accurate navigation, surveying, construction, and scientific calculations. It explains why objects disappear over the horizon, why long structures must account for curvature, and provides evidence that Earth is spherical. It's also essential for GPS, satellite communications, and modern mapping systems.