Why does the flat model work for short distances?

Over small areas the Earth's curvature is negligible, so a flat Euclidean approximation closely matches the true surface distance.

What is the Haversine formula?

It computes the great-circle distance between two points on a sphere given their latitudes and longitudes, accounting for Earth's curvature.

How large is the error at 10,000 km?

At distances around 10,000 km the flat-model error can exceed 10-15%, making it unsuitable for navigation or engineering.

Earth bulge is the height of the curved surface above the straight-line chord between two points. It matters for line-of-sight radio links and surveying.

Does this calculator use an ellipsoid model?

No, it uses a perfect sphere with R = 6,371 km. For higher accuracy, the Vincenty formula on the WGS-84 ellipsoid is needed.

Why is chord distance different from surface distance?

The chord passes straight through the Earth while the surface distance follows the curve, so the chord is always shorter.

Flat vs Round Earth Distance Calculator

Compare distances calculated using flat-earth Euclidean projection versus spherical Haversine formula to see how curvature affects real-world navigation.

Latitude 1 (°)

Longitude 1 (°)

Latitude 2 (°)

Longitude 2 (°)

Comparison Model

Distance Unit

Spherical (Haversine) Distance

3,935.75 km

Great-circle distance on a spherical Earth using the Haversine formula

Flat-Earth Distance

3,982.67 km

Euclidean distance using flat projection with latitude correction

Absolute Error

46.92 km

Difference between flat and spherical models

Relative Error

1.192%

Percentage error of the flat model compared to spherical

Central Angle

35.3950°

Angle subtended at Earth's center between the two points

Chord Distance

3,873.46 km

Straight-line distance through the Earth between the two points

Earth Bulge

301.509 km

Maximum height of the curved surface above the chord line

FlatSpherical

Distance Range	Flat Error	Spherical	Note
< 100 km	~0%	~0%	Both equivalent
100–500 km	< 0.1%	< 0.1%	Negligible difference
500–2000 km	0.1–1%	accurate	Flat starts diverging
2000–5000 km	1–5%	accurate	Significant flat error
5000–10000 km	5–15%	accurate	Flat model unreliable
> 10000 km	> 15%	accurate	Flat model breaks down

Planning notes, formulas, and examples

About the Flat vs Round Earth Distance Calculator

The shape of the Earth has a direct impact on how we measure distances between two points on its surface. For short distances — a few dozen kilometers — a flat-plane approximation works perfectly well. But as the distance grows, the curvature of the Earth introduces increasingly significant errors in the flat model.

This Flat vs Round Earth Distance Calculator lets you compare two approaches side-by-side. The spherical model uses the Haversine formula, which computes the great-circle distance along a sphere of radius 6,371 km. The flat model uses a simple Euclidean distance with a cosine latitude correction. By entering any two coordinates on Earth, you can see exactly where and how much these models diverge.

Understanding this difference is critical in aviation, maritime navigation, telecommunications (line-of-sight links), surveying, and even amateur radio. The calculator also shows the central angle subtended at Earth's center, the chord distance (straight through the Earth), and the Earth bulge — the maximum height of the curved surface above the straight-line chord. A reference table summarizes expected errors by distance range so you can quickly assess when a flat approximation is acceptable and when you must use spherical geometry.

When This Page Helps

Whether you're planning a radio link budget, studying geodesy, or simply curious about how Earth's shape affects measurements, this calculator gives you a quick side-by-side comparison with clear error metrics and visual feedback.

How to Use the Inputs

Enter the latitude and longitude of your first point (Point A).
Enter the latitude and longitude of your second point (Point B).
Select a comparison mode — both models, flat only, or spherical only.
Choose your preferred unit: kilometers, miles, or nautical miles.
Use the preset buttons for popular city pairs to load familiar route comparisons quickly.
Review the output cards for distances, error, central angle, chord distance, and Earth bulge.
Check the visual bar to see how flat distance compares to spherical distance.
Consult the reference table to understand error ranges at different scales.

Formula used

Haversine: d = 2R × arcsin(√(sin²(Δφ/2) + cos(φ₁)cos(φ₂)sin²(Δλ/2)))
Flat: d = √((Δφ × 111.32)² + (Δλ × 111.32 × cos(φ_avg))²)
Earth Bulge: b = R × (1 − cos(θ/2)) where θ is the central angle
R = 6,371 km (mean Earth radius)

Example Calculation

Result: Spherical: 3,944 km, Flat: 3,937 km, Error: 0.17%

New York to Los Angeles shows a small 0.17% error because the distance (~3,944 km) is moderate. For trans-oceanic routes the error grows well beyond 5%.

Tips & Best Practices

For distances under 100 km, the flat approximation is usually accurate to within a few meters.
Nautical miles are convenient for navigation since 1 NM ≈ 1 arc-minute of latitude.
The Earth bulge value is essential when planning microwave line-of-sight links.
Use the preset buttons to explore well-known routes without re-entering coordinates.
Remember that this calculator assumes a spherical Earth — real-world distances on the oblate ellipsoid differ by up to 0.3%.
Central angle is useful for satellite communication calculations.

Understanding Earth Models

The simplest model treats Earth as a flat plane — valid for small-scale maps like city plans. The spherical model (radius ≈ 6,371 km) captures curvature effects and is the standard for aviation and shipping. The most accurate model is the WGS-84 ellipsoid, which accounts for the equatorial bulge (~21 km wider than the polar radius).

When Does Curvature Matter?

For everyday tasks like driving directions within a city, curvature is irrelevant. For flights, maritime routes, and radio link planning beyond a few hundred kilometers, spherical geometry is essential. Engineers designing microwave relay towers must account for Earth bulge to maintain line-of-sight.

History of the Haversine Formula

The Haversine formula was published by Josef de Mendoza y Ríos in 1795 and later popularized in navigation tables. It remains the go-to formula for quick great-circle calculations because it is numerically stable even for small distances, unlike the spherical law of cosines.

Sources & Methodology

Last updated: January 15, 2025

Frequently Asked Questions

Over small areas the Earth's curvature is negligible, so a flat Euclidean approximation closely matches the true surface distance.