Great Circle Calculator

A great circle is the largest circle that can be drawn on a sphere. It is formed by the intersection of the sphere and a plane passing through its center. On Earth, the equator and all lines of longitude are great circles, but lines of latitude (except the equator) are not.

The Pythagorean theorem gives correct results only on flat surfaces. On a sphere, the shortest path curves along the surface (great circle arc), not a straight line through the interior. The Haversine formula accounts for Earth's curvature.

The Haversine formula assumes a perfect sphere with radius 6,371 km. Earth is actually an oblate spheroid, so the formula has up to 0.5% error (about 0.5 km per 100 km). For most applications, this is sufficient. For surveying-grade accuracy, use Vincenty's formula.

A great circle is the shortest path but changes bearing constantly. A rhumb line (loxodrome) follows a constant compass bearing but is a longer path. For long distances, great circle routes save significant distance.

Multiply degrees by π/180. For example, 40.7128° = 40.7128 × π/180 ≈ 0.7105 radians. Negative values (west/south) remain negative in radians.

The standard mean radius is 6,371 km (3,958.8 mi or 3,440.1 nmi). For higher precision at specific latitudes, the WGS-84 ellipsoid provides radius values from 6,357 km (poles) to 6,378 km (equator).

The initial bearing θ = atan2(sin(—λ)cos(φ₂), cos(φ₁)sin(φ₂) — sin(φ₁)cos(φ₂)cos(—λ)). Convert the result from radians to degrees and normalize to 0°—360° (add 360° if negative).

Yes, by changing the radius R. For Mars, use R ≈ 3,389.5 km. For the Moon, R ≈ 1,737.4 km. For any spherical body, substitute the appropriate radius into the Haversine formula.

Flat maps (like Mercator projection) distort the sphere onto a plane. Great circles appear as curves on most projections. On a Mercator map, a great circle route looks curved, while a straight line (rhumb line) is actually longer.

The maximum distance is half the circumference: π × R ≈ 20,004 km ≈ 12,427 miles. This occurs between antipodal points (directly opposite on the sphere, like the North and South Poles).

Airlines plan routes along great circle paths because they are the shortest distance, saving fuel and time. However, they also consider jet streams, weather, political airspace restrictions, and emergency diversion airports.

One nautical mile (nmi) equals exactly 1,852 meters and corresponds to one arcminute of latitude. It is the standard unit in navigation. One nmi ≈ 1.151 statute miles ≈ 1.852 km.

The standard formula uses Earth's mean radius. At altitude h above the surface, the effective radius becomes R + h. For aircraft at 35,000 ft (10.7 km), the correction is about 0.17%, which is usually negligible.

Yes. Routes between cities at similar longitudes but different hemispheres (e.g., New York to Bogotá) often pass near or over the North/South Pole. This is why polar routes are common for transpolar flights.

When the longitude difference exceeds 180°, subtract from 360° to get the shorter arc. For example, if —λ = 350°, use —λ = 360° — 350° = 10° instead, as the shorter path crosses the antimeridian.

For points on the equator, the great circle follows the equator itself. The distance is simply —λ × R, where —λ is the longitude difference in radians. For example, 90° along the equator ≈ 10,008 km.

Decimal degrees with 4-5 decimal places provide accuracy to about 1 meter (0.00001° ≈ 1.1 m). For most distance calculations, 2 decimal places (≈ 1 km accuracy) is sufficient. GPS coordinates typically have 6-8 decimal places.

Yes, the great circle is always the shortest path on a sphere. The only exception is when both paths have the same length, which happens when the rhumb line follows a great circle (e.g., along the equator or a meridian).

For higher accuracy on Earth's oblate spheroid, use Vincenty's formulae or the Karney algorithm. These account for Earth's equatorial bulge and provide accuracy to within millimeters. The Haversine formula is sufficient for most purposes.

The Great Circle Calculator uses standard mathematical formulas validated against authoritative references. The specific formula is displayed in the calculator interface with a detailed explanation of each variable.

Each calculator includes a step-by-step worked example showing exactly how the formula is applied. You can follow these steps with pen and paper to verify any result.

The Great Circle Calculator accepts numeric inputs including integers and decimals. Invalid inputs (letters, special characters) are rejected with clear error messages.