Enrichment · Passion Projects · Free interactive

Every flat map
gets something wrong.

Switch between six projections of the same world, watch equal-size circles swell or squash, and drag a country across the map to see its true size. All drawn in your browser from the projection formulas.

Open the explorerThe math

See distortion instead of being told about it

A sphere cannot be flattened without stretching. Each projection chooses what to keep true (angles, areas or distances from one point) and pays for it somewhere else. This explorer shows where, with numbers you can check.

Open the explorer

Space: Enrichment · Passion ProjectsAvailable

Outcome
Hands-on comparisons of six projections with measured area and angle distortion at any point, true-size country comparisons, and a map you can download as SVG or print.
Who it is for
Students in grades 6 to 12 in geography, math or a passion project, and teachers who want a live demonstration for a lesson on maps.
Format
Interactive browser tool. About 15 minutes to try every projection and two true-size drags. Touch, mouse and keyboard.
Access
Free. No account. No map services: the outlines are built into the page.

The explorer

Pick a projection. Hover or tap the map to measure. Add a country and drag it toward the poles.

Loading the tool. If nothing appears, turn on JavaScript or try another browser.

The math

Each projection is a pair of formulas from longitude λ and latitude φ to x and y, written for this page on a sphere of radius 1:

  • Mercator: x = λ, y = ln tan(π/4 + φ/2).
  • Gall–Peters (cylindrical equal-area, standard parallels 45°): x = λ cos 45°, y = sin φ / cos 45°.
  • Mollweide: solve 2θ + sin 2θ = π sin φ by Newton's method, then x = (2√2/π) λ cos θ, y = √2 sin θ.
  • Robinson: x = 0.8487 X(φ) λ, y = 1.3523 Y(φ), with X and Y from Robinson's table every 5°, interpolated.
  • Azimuthal equidistant (north polar): r = π/2 − φ, x = r sin λ, y = −r cos λ.
  • Equirectangular: x = λ, y = φ.

Measuring distortion. The probe takes the derivatives of x and y at a point numerically, divides the east-west ones by cos φ, and finds the two principal scales a and b. Area scale is a × b, shown relative to the map center; the largest angle error is 2 arcsin((a − b)/(a + b)). The page's check confirms known values: Mercator's area scale at 60° is 4 with no angle error, the equal-area maps keep a constant area scale, and Gall–Peters' angle error at the equator is about 38.9°.

Circles and true size. Each circle is every point within 6° of arc (about 667 km) of its center, so on a globe they would all match. A dragged country is turned on the sphere around the axis between its old center and the new one, which keeps its shape and size on the globe; only the projection changes how it looks. Country areas are computed from the simplified outline and are approximate.

Sources

  1. Snyder, J. P. (1987). Map Projections: A Working Manual. U.S. Geological Survey Professional Paper 1395. Formulas for Mercator, cylindrical equal-area, Mollweide and azimuthal equidistant, and the Tissot indicatrix.
  2. Snyder, J. P., and Voxland, P. M. (1989). An Album of Map Projections. U.S. Geological Survey Professional Paper 1453. Robinson's table of X and Y.
  3. Natural Earth, 1:110m physical land and admin-0 countries (public domain), naturalearthdata.com. Outlines simplified to 0.1° for this page.
  4. Bostock, M., and contributors. d3-geo-projection (ISC license): its Robinson interpolation was used as a cross-check.

Made with Natural Earth. Free vector and raster map data at naturalearthdata.com.

Questions, a problem with the tool, or a correction: team@caveatacademic.com

Outlines: Natural Earth 1:110m, public domain, simplified; country areas are approximate. Projection code is original to this page. Free to use; nothing you type leaves this browser. Caveat Academic Services · team@caveatacademic.com