Meridians of Longitude

A meridian is what bounds longitude. Where a parallel of latitude is a circle of constant latitude, a meridian is a half-circle of constant longitude — the locus of all points that share a single longitude value, from one pole to the other. The /learn/the-prime-meridian pillar covers the special meridian at 0° longitude; this support covers the rest of the family. Together with the /learn/parallels-of-latitude support and the /learn/the-equator pillar, it completes the geometric scaffolding of the geographic coordinate system.

Definition and geometry

A meridian is the intersection of Earth's surface with a plane that contains the rotation axis. Because every such plane passes through both geographic poles, every meridian runs from pole to pole. The meridian at longitude λ is the locus of all surface points whose longitude equals λ; together, the meridian at λ and the meridian at λ ± 180° (its antimeridian) form a single great circle dividing Earth into hemispheres.

A single meridian alone is a half of that great circle — a great-circle arc — covering 180° of arc length from one pole to the other. This is a basic difference from parallels of latitude: only the equator is itself a great circle; every other parallel is a small circle smaller than the equator. With meridians, every one is a great-circle arc, and there is no exception.

Per the NGA WGS 84 specification, the meridian arc length from pole to pole on the WGS 84 ellipsoid is

M = ∫₀^(π/2) ρ(φ) dφ × 2

where ρ(φ) is the meridional radius of curvature. Evaluated for WGS 84, M ≈ 20,003,931.5 m, or about 20,003.93 km — half the meridional circumference of 40,007.86 km. Every meridian has this same length because the integral is independent of which meridian you choose; the ellipsoid is rotationally symmetric about the polar axis.

The length of one degree of longitude

Where one degree of latitude is roughly constant (~111 km, varying by about 1%), one degree of longitude varies dramatically with latitude. At the equator one degree of longitude is about 111.319 km — slightly longer than one degree of latitude, because at the equator the meridian arc length per degree is smaller than the equatorial arc length per degree by about 0.1% due to ellipsoidal flattening. Away from the equator the length shrinks rapidly.

The relationship on a sphere of radius a is straightforward:

length of 1° longitude at latitude φ  =  (π / 180) × a × cos φ

At cos 0° = 1 the length is maximal; at cos 90° = 0 it vanishes. The ellipsoidal correction is small:

length of 1° longitude at latitude φ
  = (π / 180) × (a × cos φ) / √(1 − e² sin² φ)

where e² ≈ 0.00669438 is the WGS 84 first eccentricity squared.

| Latitude | Length of 1° longitude (km) | |---|---| | 0° | 111.320 | | 15° | 107.551 | | 30° | 96.486 | | 45° | 78.847 | | 60° | 55.800 | | 75° | 28.902 | | 89° | 1.945 | | 90° | 0.000 |

The practical consequence: tiles, grids, and spatial indexes that treat 1° of longitude as a fixed distance — common in poorly written geospatial code — are wrong by a factor of two by 60° latitude and by two orders of magnitude near the poles. This is a routine source of bugs in distance-threshold queries, bounding-box searches, and grid-based aggregation in latitude-spanning datasets.

A worked example: a 0.01°-square tile at the equator covers about 1.24 km², the same tile at 45° latitude covers about 0.87 km², and at 70° about 0.42 km² — a third of the equatorial area for the same longitude span. Aggregating sensor data into degree-based bins without correcting for this skew systematically under-counts polar regions. Properly implemented spatial indexes either use equal-area projections internally (S2, H3) or carry a latitude-dependent correction factor.

Why meridians converge at the poles

The convergence of meridians toward each pole is a direct consequence of the geometric definition: every meridian lies in a plane through the rotation axis. All those planes meet along the axis, and the axis pierces the surface at the two poles. The meridians are the surface traces of those planes, so they converge at the poles where the planes intersect.

The convergence is gradual at first and accelerates rapidly near the poles. Two meridians one degree apart are about 111 km apart at the equator, about 96 km apart at 30°, 79 km at 45°, 56 km at 60°, and only 1.9 km at 89°. By 89.999° they are only 1.9 metres apart. At the pole exactly, they coincide; every meridian passes through the pole, and so longitude is undefined there.

The undefined longitude at each pole is a real software gotcha. A coordinate of (90.0, 0.0) refers to the same point as (90.0, 90.0) or (90.0, −150.0) — all are the North Pole. Geocoding libraries and spatial indexes that treat coordinates as a regular two-dimensional grid produce duplicates at the poles unless they handle the singularity explicitly. Algorithms that work in the projected plane (Web Mercator clips at ±85.05°, partly to avoid this issue) sidestep the problem; those that work in geodetic coordinates have to special-case it.

Local meridians and solar time

Every point on Earth defines a local meridian — the half-circle from pole to pole that passes through it. The Sun crosses the local meridian once per solar day, the moment called local apparent noon (LAN). Per the US Naval Observatory's astronomical applications, LAN at the prime meridian corresponds (averaging over the equation of time) to 12:00 GMT.

This relationship between meridians and solar time is the historical foundation of timekeeping. Earth rotates eastward at 360° per sidereal day, or about 15° of longitude per hour. The Sun appears to cross successive meridians at 4 minutes per degree of longitude. Greenwich Mean Time was defined as the mean solar time at the Greenwich meridian — the time at which the Sun would cross the meridian averaged over a year.

Civil time today is decoupled from this astronomical anchor: every time zone uses a single offset from UTC regardless of the observer's exact longitude. A clock in Madrid (3.7°W of the prime meridian) is set the same as one in Warsaw (21°E), even though the local meridian transit of the Sun differs by about 100 minutes between them. The /learn/time-zones-explained pillar covers the political compromises that decouple civil time from solar time.

Meridians on map projections

Together, parallels and meridians form the graticule — the two-dimensional net of intersecting lines that organises every map. Different projections draw the graticule differently, and the meridian treatment is one of the most visible choices.

• Mercator (cylindrical, conformal): meridians are equally spaced vertical lines. The projection preserves angles and is straightforward for navigation by rhumb line, at the cost of severe area distortion toward the poles.
• Web Mercator (EPSG:3857, used by every web map tile): same meridian treatment as Mercator, but using a sphere rather than the ellipsoid. The /learn/web-mercator-projection support covers the simplification.
• Lambert conformal conic (used by FAA aviation charts): meridians are straight lines converging at a single point above (or below) the map. Designed for mid-latitude regions of east-west extent.
• Robinson (used by National Geographic world maps): meridians are smooth curves; no single property is preserved, but distortion is visually distributed.
• Polar stereographic (above 80° latitude): meridians are straight lines radiating from a central pole.

The /learn/what-is-a-map-projection pillar covers the impossibility theorem (Gauss's Theorema Egregium) that forces every projection to distort something; how the meridians are drawn is one signature of which property the projection chose to keep.

Meridian arcs and the shape of Earth

Before satellites, the only way to measure the Earth's shape was to measure long arcs of meridian on the ground and compare the corresponding latitude differences. If Earth were a perfect sphere, one degree of meridian arc would have the same length everywhere; if Earth were oblate (wider at the equator), one degree near the poles would be longer than one degree near the equator, because the radius of curvature at the poles is larger.

Two famous meridian-arc expeditions resolved Earth's shape in the 18th century:

• The French Geodesic Mission to Quito (1735–1744) — Bouguer, La Condamine, and Godin measured a meridian arc at the equator in the Andes of present-day Ecuador.
• The French Geodesic Mission to Lapland (1736–1737) — Maupertuis and Clairaut measured a meridian arc at high northern latitude.

Comparing the two expeditions' arcs confirmed Newton's prediction that one degree near the poles is slightly longer — Earth is oblate. The /learn/why-the-earth-is-not-a-sphere support covers the physics; the /learn/john-harrison-and-the-marine-chronometer support covers the longitude problem that needed independent solving in the same era.

The meridian arc remained the gold-standard tool for defining the shape of Earth into the 20th century. The metre was defined in 1791 as one ten-millionth of the meridian quadrant from Paris to the North Pole, measured by Delambre and Méchain between 1792 and 1799 along the Dunkirk-to-Barcelona arc. The modern metre is defined by the SI second and the speed of light, but the original definition lives on as the reason 40,000 km is roughly Earth's meridional circumference — that relationship was designed, not coincidence.

Antimeridians and the date line

The antimeridian of meridian λ is the meridian at λ ± 180°. Each meridian and its antimeridian together form a great circle. The prime meridian and the 180th meridian form the longitude reference great circle.

The 180th meridian — also called the antimeridian without further qualification — is geometrically interesting because longitudes +180° and −180° describe the same line. Software that handles geometry must split polygons that cross the antimeridian and handle the edge case of values that wrap past it; the /learn/the-international-date-line support covers the political line that broadly follows the 180th meridian with deviations to keep individual countries on a single calendar day.

Sources

• NGA, World Geodetic System 1984 — ellipsoid parameters and meridian-arc length.
• Snyder, Map Projections — A Working Manual (USGS Prof. Paper 1395) — meridional radius of curvature and the projection treatment of meridians.
• NOAA NGS, Datums and Reference Frames — the reference frame context.
• BIPM, Coordinated Universal Time — civil-time convention based on the prime meridian.
• US Naval Observatory, Astronomical Applications — solar time, local apparent noon, and the equation of time.

For the historical context of meridian-arc measurements, see /learn/the-1884-international-meridian-conference and /learn/why-the-earth-is-not-a-sphere; for the prime meridian specifically, see /learn/the-prime-meridian.