Cartographic Projection
every map is wrong, and by exactly how much
What is taught wrongly

The projection that shows true size

There is no such thing, and the phrase hides a real question. Equal-area projections preserve area and destroy shape; nothing preserves size in the sense the phrase implies; and the tools that make the point best are not maps at all.
13 min read 6 figures Purpose before propertyThe trade-off is forcedMeasured, not named

“A map that shows the true size of countries” is a phrase that appears constantly, usually attached to an equal-area world map. It describes something people genuinely want and no projection provides.

One 20° × 10° cell at 60°, under four projectionsThe same patch of the sphere, drawn by four projections and scaled to fit. The shapes differ and so do the areas: the figure under each panel is the areal inflation relative to the same cell at the equator, so 1.00 means the projection treats the two fairly.Mercator5.62× areaGall–Peters1.00× areaMollweide1.00× areaEquirectangular2.36× areathe cell at 60°scaled to fit, areas as stated
Fig. 1 The same 20°-by-10° patch of sphere at 60–70° north, under four projections, each scaled to fit. Two of them get the area exactly right. None of them gets the shape right, and the four disagree about what the patch even looks like.

What equal-area actually guarantees

Precisely one thing: that the ratio of any region’s area on the map to its area on the sphere is the same everywhere. Measure two countries on a Gall–Peters map, take the ratio, and it equals the true ratio exactly.

That is a real and useful guarantee. This site verifies it — the equal-area projections measure their areal error between 6×10−126\times10^{-12} and 1.5×10−91.5\times10^{-9}, which is arithmetic noise.

What it does not guarantee is that a country looks the right size, and the gap between those two statements is what the phrase “true size” papers over.

Area is a single number. Apparent size is a matter of shape, extent, and how the eye integrates a form — and an equal-area projection is free to distort shape as much as it likes, which it must, because keeping both would be an isometry.

A country stretched to twice its height and half its width has the same area and does not look the same size. On Gall–Peters near the equator that is roughly what happens.

The measured shape cost

Tissot's indicatrix across Gall–PetersA small circle on the sphere, drawn where the projection puts it. The dashed circle behind each is what an undistorted map would show. On Gall–Peters ω reaches 39°, and the areal factor reaches 1.0.dashed: an undistorted circledrawn in Gall–Peters
Fig. 2 Gall–Peters with its indicatrices. Every ellipse has the same area — that is the guarantee, and it holds exactly. None is a circle except at 45°, and the measured maximum angular deformation is about 146°.

146 degrees is severe. Shapes near the equator are stretched vertically by a factor of two and shapes near the poles are flattened horizontally by a comparable amount, and continents are visibly wrong.

So an equal-area world map trades a distortion that is invisible and large for one that is visible and large. Which of those is preferable depends entirely on the map’s purpose, and for a thematic map of anything measured per unit area it is clearly the right trade.

For “showing the true size of countries”, it is a strange one — because it fixes the number and breaks the picture, and the phrase is about the picture.

What the phrase is actually after

Two different things, usually conflated.

The relative areas, as numbers. Africa is about fourteen times Greenland. This is a fact about the world and any equal-area map preserves it; so does a table, and a table preserves it better because reading areas off a map by eye is unreliable regardless of the projection.

A sense of relative size, visually. This is what the phrase means and no world map delivers it, because seeing that one shape is fourteen times another requires comparing them side by side at the same place on the map, which a world map by definition does not do.

That second point is the crux. Even on a perfect equal-area map, Greenland and Africa are far apart, differently shaped, and at different latitudes with different local shape distortions. The comparison the phrase wants is not a comparison a map can present.

What actually works

The tools that make the point convincingly abandon the projection.

The well-known ones let a country’s outline be dragged across a map, rescaling it as it moves so that its true area is preserved at each new latitude. Watching Greenland shrink as it slides toward the equator on a Mercator map is far more persuasive than any static equal-area map, and it works because it is doing the comparison directly: the same shape, in two places, at the same true size.

That is not a projection. It is an interaction that compensates for a projection, and it succeeds precisely because it sidesteps the problem rather than solving it.

The other thing that works is a globe. A globe has no distortion at all, which is the only way to have none, and its cost is that half of it is facing away at any moment. Every flat figure on this site is a projection, including the ones that look like globes — an orthographic view is a projection with its own distortion, showing a hemisphere, flattening badly toward the rim.

Five identical cells on Gall–PetersFive patches, each 20° of longitude by 10° of latitude. On the sphere the higher ones are genuinely smaller, because the meridians converge. On Gall–Peters the cell at 70° comes out 1.0 times larger than the equatorial one relative to its true size.1.0×1.0×1.0×1.0×1.0×each cell is 20° × 10°drawn in Gall–Peters
Fig. 3 The comparison cells on Gall–Peters. Each patch is 20° by 10°, and each carries its measured areal inflation relative to the equatorial one. All five read 1.0, exactly, which is what equal-area means — and the shapes are visibly different from one another all the same.

That figure is the essay in one picture. The numbers are all 1.0 and the shapes disagree, and both facts are the projection working correctly.

Where the phrase does real harm

Not in casual use, where it is harmless shorthand. In two specific places.

Thematic mapping. A choropleth of population density on a conformal projection is genuinely misleading, because the visual weight of each region is proportional to its inflated area rather than to the data. Here “true size” points at a real requirement and equal-area is the right answer — and the requirement is about the data being read correctly, not about the countries looking right.

Claims of neutrality. Presenting an equal-area map as the undistorted one, or the honest one, misdescribes what it is. It is a map that has chosen area over shape, which is a choice with reasons, and calling the choice an absence of choice is the part that will not survive contact with the measurements.

The honest version of the phrase

Something like: a map on which the ratio of any two regions’ areas is correct.

Longer, less quotable, and true. It states the guarantee, implies the cost by not mentioning shape, and does not suggest that any map shows anything the way it is.

The general form of this is worth carrying elsewhere: a projection preserves a named property, and any description of a map that does not name the property is describing an impression.

What the phrase does correctly identify

Being fair to it: there is a genuine failure mode the phrase is reacting to, and dismissing it entirely would be wrong.

A world map on which Greenland appears comparable in size to Africa is conveying something false, and a reader who has only ever seen such maps will have a systematically wrong sense of global proportion. That is a real problem with a real cause, and the cause is measurable at fifteenfold.

What the phrase gets wrong is the remedy. The fix is not a projection that shows true size, because there is none; it is either an equal-area projection with its shape cost stated, or a comparison tool, or a globe — and choosing among those depends on what the map is for.

Five identical cells on MollweideFive patches, each 20° of longitude by 10° of latitude. On the sphere the higher ones are genuinely smaller, because the meridians converge. On Mollweide the cell at 70° comes out 1.0 times larger than the equatorial one relative to its true size.1.0×1.0×1.0×1.0×1.0×each cell is 20° × 10°drawn in Mollweide
Fig. 4 The comparison cells on Mollweide, which is equal-area with much gentler shape distortion than Gall–Peters. Every cell reads 1.0, exactly. If an equal-area world map is wanted, this is a better one — and the argument was conducted almost entirely about a different projection.

The same phrase in other subjects

The pattern is worth recognising because it recurs wherever a representation has to choose.

A statistical graphic that “shows the data as it really is” has chosen a scale, an aggregation and an aspect ratio. A photograph that “shows what it really looked like” has chosen an exposure, a white balance and a focal length. In each case the phrase claims an absence of choice, and in each case the choice is unavoidable and consequential.

The honest form is always the same: name what has been preserved. A map that preserves area ratios; a chart on a logarithmic scale; a photograph white-balanced to daylight. Longer, less quotable, and checkable.

The measurement that settles it

The simplest way to see that equal-area does not mean true size is to put the two claims side by side on one figure.

Every cell in the equal-area panels reads exactly 1.0 — the areal guarantee, holding to a part in 101110^{11}. And the cells are visibly different shapes from one another, because the shape distortion varies with latitude and is not constrained by anything.

A reader looking at those panels and asked which patch is largest would give different answers depending on the panel, and would be wrong in every case except by accident. The number is right and the impression is not, and the phrase “true size” is about the impression.

One 20° × 10° cell at 45°, under four projectionsThe same patch of the sphere, drawn by four projections and scaled to fit. The shapes differ and so do the areas: the figure under each panel is the areal inflation relative to the same cell at the equator, so 1.00 means the projection treats the two fairly.Mercator2.41× areaGall–Peters1.00× areaMollweide1.00× areaEquirectangular1.55× areathe cell at 45°scaled to fit, areas as stated
Fig. 5 A 20°-by-10° cell at 45° under four projections, scaled to fit. Two of the four preserve its area exactly. All four disagree about its shape, and none of them conveys its size, because each panel has been scaled independently.

What a cartogram does

There is one more family worth mentioning, because it takes the phrase seriously and abandons geography to do it.

A cartogram distorts the map so that each region’s area represents some quantity — population, GDP, votes — rather than its geographic extent. Shapes are abandoned entirely, and often so is contiguity.

That is the logical endpoint of prioritising a quantity over appearance, and it is honest about what it has done: nobody looks at a population cartogram and thinks it is a picture of the coastline.

Which is the contrast worth drawing. A cartogram sacrifices shape openly for a stated purpose. An equal-area world map sacrifices shape for a stated purpose too, and is sometimes presented as though it had sacrificed nothing.

Tissot's indicatrix across HammerA small circle on the sphere, drawn where the projection puts it. The dashed circle behind each is what an undistorted map would show. On Hammer ω reaches 80°, and the areal factor reaches 1.0.dashed: an undistorted circledrawn in Hammer
Fig. 6 Hammer’s indicatrices, equal-area with far gentler shape distortion than Gall–Peters. If an equal-area world map is wanted, projections like this one exist and were available throughout the argument about a different one.

A phrase worth retiring

The suggestion this essay ends on is narrow: replace “true size” with “equal-area”, and say what it costs.

“Equal-area” is a technical term with an exact meaning, it is checkable, and it does not imply that anything else about the map is true. It is also two words, so nothing is lost in brevity.

What is lost is the rhetorical force, and that is the point. “True size” persuades by implying an absence of choice, and there is no absence of choice available — so the persuasion is doing work the geometry does not support, and a reader convinced by it has been convinced of something false alongside something true.

The broader habit this essay recommends is to be suspicious of any description of a representation that claims to have made no choice. A map, a chart, a photograph and a summary statistic all discard something, and the useful question is always which thing — asked of the representation rather than of the person presenting it.

One last note on what the site can and cannot settle here. That equal-area projections preserve area is measured, exactly. That Gall-Peters distorts shape by 146 degrees is measured. Whether a reader’s sense of the world improves more from an equal-area map than from a compromise is not measurable by anything here, and it is the question the phrase is really about.

A final practical note. Anyone wanting to convey relative sizes has better options than any world map: a bar chart, a table, a treemap, or a direct side-by-side overlay. All of them beat a map at that specific job, and the reason maps are used instead is that the map is also doing something else — showing where things are — which is the requirement no chart satisfies.

The phrase also has a useful diagnostic property. Somebody who says a map shows true size is usually reasoning about area and unaware that shape was traded for it, which means the conversation worth having is about the two independent distortions rather than about which projection to prefer.

The phrase’s persistence is also a reminder that a good correction can carry a bad claim. The observation that Mercator misrepresents relative size is correct, important and worth repeating; the packaging that presents an equal-area map as showing things as they are is not; and the two travel together because the second makes the first more quotable.

What was computed here

Every areal figure comes from the closed-form area of a latitude–longitude cell, R2 Δλ (sin⁡φ2−sin⁡φ1)R^2\,\Delta\lambda\,(\sin\varphi_2 - \sin\varphi_1), which is exact and involves no dataset. The formula is verified by summing a full covering of the sphere and requiring 4π4\pi, to 7×10−157\times10^{-15}.

The equal-area projections are asserted to pass the area test and to fail the conformality test on every build, and both halves matter — the second is what makes the shape-cost claim in this essay a measurement rather than an impression.

The site deliberately does not compute the Greenland-to-Africa ratio from coastline polygons, because a polygon’s area depends on its simplification level and the figure would be partly a measurement of a vendor’s generalisation. The ratio is quoted as the geographic measurement it is; the mechanism is computed exactly.

What the pictures cannot show

Apparent size, which is the whole subject. The cell figures scale each panel to fit, and scaling is exactly the operation that destroys an areal comparison — so the numbers are printed because the drawing cannot carry them.

Drawing four differently-sized shapes at true relative size would mean three of them being too small to see. That is not a limitation of this site’s figures; it is why the phrase “shows true size” is asking for something a single static image cannot give.

The essay also cannot show the interactive comparison that does work, for the same reason it works: it needs to be operated.

Who found it, and when

The equal-area cylindrical projection is Lambert’s, from 1772, and he named the property as the design goal. Gall produced the 45° variant in 1855 and Peters re-presented it in 1973, which is where the modern argument starts.

The framing of equal-area as the true projection is largely a product of that argument and its afterlife, and it is not a claim Gall or Lambert made. Lambert in particular constructed conformal and equal-area projections side by side in the same publication, naming both properties as goals — which is a fair indication that he understood them as alternatives rather than as truth and error.

The interactive size-comparison tools date from around 2013 and have probably done more to correct public intuition about Mercator than the whole preceding forty years of argument, by declining to argue and showing the comparison instead.

Where this goes next

The argument this essay sits inside is Mercator against Peters. The theorem that makes the trade unavoidable is the trade-off is two lines. And for what a map with no exact property can offer instead, compromise projections.

Where this goes next

equator23°45°60°70°1×MercatorEquirectangularMillerGall–Peterslatitude of the cell1× means treated fairlyrelative to the equator Mercator against Peters The most-argued question in cartography, conducted almost entirely without anyone measuring anything. Both projections are exactly what they claim, each destroys what the other keeps, and the numbers are computable in either direction. 10⁻⁶10⁻⁴10⁻²110010⁻¹¹10⁻⁸10⁻⁵10⁻²1010000maximum angular deformation / degreesmaximum areal errornothing here, everMercatorStereographicWeb MercatorGall–PetersMollweideEquirectangulartolerances: 0.0001° and 0.00000121 projections measured The trade-off is two lines Conformal means the two principal scales are equal. Equal-area means their product is one. Both at once forces both to one, which is an isometry, which the curvature forbids. That is the entire argument. RobinsonWinkel tripelMollweideMercatorGall–PetersEckert IVsame sphere, same graticuleno two agree Compromise projections A projection that preserves nothing exactly can distort everything less than one that preserves something exactly. For a general-purpose world map that is the right trade, and it is why the two most widely used ones today have no exact property at all.