What is a great circle?

The shortest path between two points on a sphere—the circle that divides the sphere into two equal hemispheres. Geodesic on the sphere.

How is great circle distance computed?

Central angle Δσ is found using the Spherical Law of Cosines from the two lat/lon points; distance d = R × Δσ (R = sphere radius).

What is Earth's mean radius?

About 6371 km. Used as default for Earth; equatorial and polar radii differ slightly.

Why do flight paths look curved on a map?

They follow great circles—the shortest path. Flat maps distort; the great circle projects as a curve except along equator or meridian.

What is initial vs final bearing?

Initial bearing is the compass direction at the start; final bearing at the destination. They differ except when traveling along equator or a meridian.

Does this work for other planets?

Yes. Use the sphere radius (e.g. Mars ~3390 km, Moon ~1737 km) and lat/lon of the two points.

5 more

GEOMETRY3D GeometryMathematics Calculator

🌍

Great Circle — Shortest Path on a Sphere

Compute the geodesic distance between two points on a sphere, plus initial/final bearings and midpoint. Essential for aviation, maritime navigation, and GPS.

Concept Fundamentals

d = R × Δσ

Distance

Spherical Law of Cosines

Central Angle

~6371 km

Earth Radius

0°=N, 90°=E

Bearings

Did our AI summary help? Let us know.

Great circle = geodesic = shortest surface path Flight paths follow great circles—they appear curved on flat maps Initial bearing ≠ final bearing except along equator or meridian Earth mean radius ~6371 km gives ~0.5% typical error Works for any sphere—Mars, Moon, or custom radius

Key quantities

d = R × Δσ

Distance

Key relation

Spherical Law of Cosines

Central Angle

Key relation

~6371 km

Earth Radius

Key relation

0°=N, 90°=E

Bearings

Key relation

Ready to run the numbers?

Why: Great circles define the shortest path between two points on a sphere. Flight paths, shipping routes, and GPS calculations all use spherical geometry. Understanding geodesics is fundamental to navigation and geospatial applications.

How: Enter latitude and longitude for two points, plus the sphere radius (6371 km for Earth). The calculator uses the Spherical Law of Cosines for the central angle, then multiplies by radius for distance. Bearings and midpoint use standard spherical trigonometry formulas.

Great circle = geodesic = shortest surface pathFlight paths follow great circles—they appear curved on flat maps

Sources:Wikipedia — Great-circle distanceWolfram — Spherical Trigonometry

Run the calculator when you are ready.

Great Circle CalculatorEnter two lat/lon points and sphere radius to compute distance, bearings, and midpoint

🌍

3D GEOMETRYGeodesic

Great Circle — Shortest Path on a Sphere

Distance, bearings, and midpoint between two points. Essential for navigation.

🌍 Sample Routes — Click to Load

Coordinates & Radius

Point 1

Latitude (°)

Longitude (°)

Point 2

Latitude (°)

Longitude (°)

Sphere Radius

Unit

greatcircle.sh

CALCULATED

$ calc --lat1=40.71 --lon1=-74.01 --lat2=51.47 --lon2=-0.45

Distance

5,550.8 km

Central Angle

49.92°

Initial Bearing

51.3°

Final Bearing

108.2°

Midpoint Lat

52.2927°

Midpoint Lon

-41.4127°

Great Circle Distance

5,551 km

Bearing: 51.3° → 108.2°

numbervibe.com/calculators/mathematics/3d-geometry/great-circle-calculator

Visualization

Property Radar

Property Comparison

Property Breakdown

📐 Calculation Breakdown

INPUT

Point 1

(40.71°, -74.01°)

Point 2

(51.47°, -0.45°)

RESULTS

Great Circle Distance

5,550.7951 km

d = R imes \text{Delta} \text{sigma}

Central Angle

49.9195°

BEARINGS

Initial Bearing

51.34°

Final Bearing

108.16°

MIDPOINT

Midpoint

(52.2927°, -41.4127°)

For educational and informational purposes only. Verify with a qualified professional.

🧮 Fascinating Math Facts

✈️

Flight paths follow great circles—routes appear curved on flat maps

— Aviation

🌍

Earth mean radius ~6371 km; equatorial and polar radii differ slightly

— Geodesy

🧭

Initial bearing ≠ final bearing except when traveling along equator or meridian

— Navigation

📐

Haversine formula is an alternative, more stable for small distances

— Spherical Geometry

📋 Key Takeaways

• Great circle = shortest path on a sphere (geodesic)
• Distance d = R × Δσ (radius × central angle in radians)
• Uses Spherical Law of Cosines for central angle
• Bearings measured from North (0°=N, 90°=E)

💡 Did You Know?

✈️Flight paths follow great circles — routes appear curved on flat maps

🌍Earth mean radius ~6371 km; equatorial and polar radii differ slightly

🧭Initial bearing ≠ final bearing except when traveling along equator or meridian

📐Haversine formula is an alternative, more stable for small distances

🗺️GPS and mapping software use great circle calculations internally

🚢Maritime and aviation navigation rely on great circle routes for fuel efficiency

🛰️Satellite orbits and communication paths use spherical geometry

📖 How It Works

Central Angle

Δσ = arccos(sin φ₁ sin φ₂ + cos φ₁ cos φ₂ cos Δλ) — Spherical Law of Cosines.

Distance

d = R × Δσ (radians). Same unit as radius.

Bearings & Midpoint

Initial/final bearings from atan2 formulas. Midpoint via spherical interpolation.

🎯 Expert Tips

Decimal Degrees

Use decimal degrees (e.g. 40.71) not DMS. North/East positive.

Radius Units

Distance output uses same unit as radius (km, mi, nmi).

Other Spheres

Use Mars radius (~3390 km), Moon (~1737 km) for other bodies.

Accuracy

Earth is oblate; mean radius gives ~0.5% error for most routes.

⚖️ Comparison

Method	Use Case
Spherical Law of Cosines	General, used here
Haversine	Small distances, numeric stability
Vincenty	Ellipsoid (WGS84), highest accuracy

❓ FAQ

Great circle vs straight line?

Great circle is shortest path on surface. Straight line through Earth is shorter but not a surface path.

Why do flight paths look curved?

Flat maps distort. Great circles project as curves on Mercator.

Does point order matter?

No. Distance is same. Bearings swap (init↔final).

Other planets?

Yes. Enter coordinates and that body's radius.

How accurate is this?

Spherical model: ~0.5% error. For survey-grade use ellipsoid (Vincenty).

What is central angle?

Angle at sphere center between the two points. d = R × Δσ.

📊 Stats

6371

Earth radius (km)

~0.5%

Typical error

Points needed

Degrees input

📚 Sources

Wikipedia — Great-circle distance ↗

Movable Type — Formulas ↗

Wolfram — Spherical Trigonometry ↗

ICAO — Aviation Standards ↗

⚠️ Disclaimer: Uses spherical Earth. For geodetic accuracy use WGS84 ellipsoid models.

👈 START HERE

⬅️Jump in and explore the concept!

Great Circle — Shortest Path on a Sphere

Great Circle — Shortest Path on a Sphere

🌍 Sample Routes — Click to Load

Coordinates & Radius

Point 1

Point 2

Visualization

Property Radar

Property Comparison

Property Breakdown

📐 Calculation Breakdown

🧮 Fascinating Math Facts

📋 Key Takeaways

💡 Did You Know?

📖 How It Works

Central Angle

Distance

Bearings & Midpoint

🎯 Expert Tips

Decimal Degrees

Radius Units

Other Spheres

Accuracy

⚖️ Comparison

❓ FAQ

Great circle vs straight line?

Why do flight paths look curved?

Does point order matter?

Other planets?

How accurate is this?

What is central angle?

📊 Stats

📚 Sources

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