Equal-Area Projections

Equal-area projections are one of the three main projection-property families. The /learn/what-is-a-map-projection pillar covers the impossibility theorem (Gauss's Theorema Egregium) that prevents any single projection from preserving every property at once; equal-area projections sacrifice shape (and angle, and distance) to preserve area exactly. The /learn/conformal-projections support covers the opposite family — projections that preserve angles — including the /learn/mercator-projection. This article covers the equal-area family in breadth: what the property means, why it matters for specific use cases, and how to choose a particular projection within the family.

The property defined

An equal-area projection is a projection in which the areal scale factor is constant. Given any small region on Earth's surface, its image on the map has area equal to its true area multiplied by a single global constant. The constant typically depends on the scale of the map (a 1:25,000,000 world map will have a smaller constant than a 1:1,000,000 regional map), but it does not vary with latitude, longitude, or any other geographic parameter.

Mathematically, the projection function (φ, λ) → (x, y) has a Jacobian determinant

J = (∂x/∂φ)(∂y/∂λ) − (∂x/∂λ)(∂y/∂φ)

that is constant (when projecting from a unit sphere). Conformality — the property preserved by Mercator and other conformal projections — requires the Jacobian to be a scalar multiple of a rotation matrix, which is a stronger condition that no equal-area projection satisfies. The two properties are mutually exclusive: no projection can be both equal-area and conformal except the identity (which works only on a flat surface).

The geometric consequence of equal-area is that small shapes are not generally preserved. A small circle on the ellipsoid maps to an ellipse on the projection, and the ratio of the ellipse's axes varies across the map. This is exactly the Tissot indicatrix you see in cartographic textbooks; for an equal-area projection, every indicatrix has the same area, but its eccentricity (and orientation) varies with location.

Why equal-area matters

For any thematic map showing a quantity per unit area, equal-area projection is mandatory unless one is willing to distort the data:

• Population density. If a country's land area is inflated by the projection (as Russia, Canada, and Greenland are on Mercator), a population-density map underestimates how dense that country appears relative to lower-latitude countries. The Russian Far East looks empty on Mercator partly because the area is inflated; it is empty, but less dramatically so than Mercator suggests.
• Election results / political mapping. Maps showing percent-of-vote by district need equal-area to avoid visually weighting high-latitude districts.
• Climate data. Rainfall, temperature anomaly, and other gridded climate quantities are typically published per unit area; equal-area projection preserves the proportions.
• Agricultural and economic statistics. GDP per square kilometre, yield per hectare, and other economic intensity measures depend on area being preserved.

The reverse case — when shape is what matters — favours conformal projections: navigation charts, aeronautical charts, weather forecasting maps that show wind direction. The /learn/conformal-projections support covers these.

For visual reference without thematic data overlay, compromise projections like Robinson or Winkel Tripel are often preferred over either pure equal-area or pure conformal alternatives. The choice depends on the map's purpose.

The major families

Cylindrical equal-area

The simplest equal-area family: parallels are horizontal lines, meridians are equally spaced vertical lines, with vertical spacing of parallels adjusted to preserve area. Lambert proposed the first member in 1772; many specific projections in the family differ only by their standard parallel choice:

| Projection | Standard parallels | Year | Notes | |---|---|---|---| | Lambert cylindrical | 0° | 1772 | Original; severe polar stretching | | Behrmann | ±30° | 1910 | Compromise within the family | | Smyth equal-surface | ±37.5° | 1870 | Roughly square countries | | Hobo-Dyer | ±37.5° | 2002 | Modern rebranding of Smyth | | Gall-Peters | ±45° | 1855/1973 | The famous one; see Peters | | Tobler hyperelliptical | ±55° | 1973 | Polar areas correct |

All members preserve area; they differ in which latitudes look relatively undistorted.

Pseudocylindrical equal-area

Parallels are horizontal lines, but meridians curve, allowing better shape preservation:

• Sinusoidal projection (Cossin 1570, also called Mercator-Sanson): meridians are sine curves; parallels equally spaced. Severe shape distortion at the corners.
• Mollweide projection (Carl Mollweide 1805): meridians are ellipses, the map outline is an ellipse with 2:1 aspect ratio. The classic "elliptical world map".
• Hammer projection (Ernst Hammer 1892): a modified azimuthal with an elliptical outline, equal-area; popular for hemisphere-emphasis maps.
• Eckert IV (Max Eckert 1906): pseudocylindrical with elliptical meridians; somewhat similar to Mollweide but with pole lines rather than pole points. Often used in geographic textbooks.
• Eckert VI (Max Eckert 1906): similar to Eckert IV but with sinusoidal meridians.
• Wagner IV / VI / VII (Karlheinz Wagner 1932-49): a family of modifications of earlier pseudocylindrical projections.

Azimuthal equal-area

The Lambert azimuthal equal-area projection (Lambert 1772) is the canonical member. From a central point, every point on Earth's surface is mapped to a point at the correct distance (along a straight line through the centre). Equal-area variants of the azimuthal projection are widely used for:

• Continental thematic maps: ETRS89-LAEA-Europe (EPSG:3035, the European Inspire-recommended projection), Albers conic for the US, equivalent projections centred on Africa or Asia.
• Hemisphere maps: a Lambert azimuthal centred at one pole, the equator, or any other point captures roughly half of Earth's surface in a single map.
• Statistical maps: when the data is concentrated in one region, centring the projection there minimises distortion within the relevant area.

Conic equal-area

• Albers conic equal-area (Heinrich Albers 1805): the standard projection for continental US, Russia, and other east-west elongate landmasses. Uses two standard parallels (29.5°N and 45.5°N for US-wide use; other pairs for other regions). The USGS National Map has long used Albers for US-wide thematic maps.
• Bonne projection (Rigobert Bonne 1752): pseudoconic, with curved parallels and meridians. Older but still used in some national mapping series.

Interrupted equal-area

Projecting the world as a single connected shape always introduces substantial distortion somewhere. Interrupted projections cut the world into multiple pieces to reduce distortion:

• Goode homolosine projection (J. Paul Goode 1923) is the classic. The projection alternates between the sinusoidal (for low latitudes) and Mollweide (for high latitudes), and is typically presented in an interrupted form that splits the oceans into multiple lobes. Long popular in classroom atlases for thematic mapping.
• Berghaus star projection (Heinrich Berghaus 1879) — an azimuthal equidistant cut into a five-pointed star, occasionally used for political symbolism (the United Nations emblem uses a closely related design).

Modern compromise-equal-area

• Equal Earth (Bojan Šavrič, Tom Patterson, Bernhard Jenny, 2018) — explicitly designed as an equal-area projection that also looks visually balanced. Per the original IJGIS paper, the design borrows shape characteristics from the Robinson and Winkel Tripel projections while maintaining strict equal-area preservation. Rapidly being adopted as the default equal-area world projection.
• Natural Earth II (Tom Patterson 2007/2014) — Patterson's earlier project, also pseudocylindrical with attention to visual balance, but not strictly equal-area; superseded for thematic use by Equal Earth.

Practical guidance

The default choice for new design work depends on the use case:

| Use case | Recommended equal-area projection | |---|---| | US-wide thematic | Albers conic, std parallels 29.5°N and 45.5°N (USGS standard) | | US state-level | Albers conic with state-specific standard parallels | | Europe-wide thematic | ETRS89-LAEA-Europe (EPSG:3035) | | Continental Africa, Asia, Oceania | Lambert azimuthal centred on the continent | | Single-country small | Whichever national projection is equal-area (most are) | | World thematic, general purpose | Equal Earth (modern), Mollweide (classic) | | World thematic, ocean-emphasis | Goode homolosine, interrupted | | Polar regions | Lambert azimuthal equal-area centred on pole |

The choice involves trade-offs around shape preservation, familiarity, and software support. Modern cartographic libraries (PROJ, GDAL, D3.js, MapLibre) include all of these projections; the legal/published preferences (ETRS89-LAEA in Europe, Albers in the US) are partly historical and partly mandated by national mapping agencies.

Tissot indicatrix and shape distortion

The standard tool for visualising projection distortion is the Tissot indicatrix, introduced by Nicolas Auguste Tissot in 1859. The construction: draw a small circle at a point on the Earth's surface, then project that circle onto the map. On a conformal projection (Mercator, Lambert conformal conic, stereographic), the projected shape is always a circle — possibly larger or smaller than the original, but still circular. On an equal-area projection, the projected shape is an ellipse whose area equals the original circle's area, but whose axes are unequal (the ratio of semi-major to semi-minor axes measures the shape distortion).

The Tissot indicatrix lets a viewer see at a glance which projection preserves which property:

• Conformal projection. Indicatrices are circles, sizes vary with location (small at the equator on Mercator, very large at the poles). Same shape, varying size.
• Equal-area projection. Indicatrices are ellipses, all with the same area, but their eccentricity varies. Same area, varying shape.
• Compromise projection. Indicatrices are ellipses of varying area and varying eccentricity. Neither property is preserved exactly, but neither is destroyed.

The ratio between the semi-axes of the indicatrix is a useful single-number distortion measure. For Peters at 0°, the indicatrix is nearly circular but compressed vertically by about 36%; at the standard parallels of ±45°, it is exactly circular; at the poles, it is extremely elongated horizontally. The cylindrical equal-area family produces the most variable indicatrix shapes; pseudocylindrical and azimuthal equal-area projections distribute the variation more evenly.

For software that visualises distortion, the Equal Earth project and several academic cartography tools render Tissot indicatrices automatically across any projection.

A worked area-preservation example

Take Russia: total land area 17.10 million km². On Mercator, Russia appears to occupy a much larger fraction of the world map than its true area justifies. On Lambert cylindrical equal-area (the simplest cylindrical equal-area projection), Russia's map area equals the correct 17.10 million km² scaled by the global constant.

The global constant for a 1:30,000,000 Lambert cylindrical equal-area map of the world is (surface of Earth) / (map dimensions)²:

Earth surface area ≈ 510.1 million km²
At 1:30,000,000, scaled area ≈ 510.1e6 / (30e6)² = 5.67e-7 km²·km⁻²
                            ≈ 0.567 cm² per million-km² of true area
Russia's 17.10 M km² → 17.10 · 0.567 = 9.69 cm² on the 30M-scale map

This precise computation works for every equal-area projection at the same map scale, and the answer (9.69 cm² for Russia) is identical. The differences between equal-area projections are entirely in where on the map Russia appears and what shape it has, not in how much area it occupies.

Sources

• Snyder, Map Projections — A Working Manual (USGS Prof. Paper 1395) — the standard reference.
• Snyder, Flattening the Earth (University of Chicago Press, 1993) — historical context for every projection mentioned.
• Robinson, Snyder, Voxland, Album of Map Projections (USGS Bulletin 1453, 1989) — visual catalogue with sample maps.
• Šavrič, Patterson, Jenny, The Equal Earth map projection (IJGIS, 2018) — Equal Earth design paper.
• NASA Earth Observatory, Map Projections: The Good, the Bad, and the Ugly — accessible visual comparison.

For closely related material, see /learn/peters-projection for the most famous equal-area projection, /learn/conformal-projections for the opposite property, and /learn/what-is-a-map-projection for the impossibility theorem that forces every projection into a trade-off.