Choose a projection for the question

Choose a projection for the question

Question and evidence

Can a visually larger country have a smaller measured area?

A fictional atlas contains two maps of the same countries. Its area table lists North at 2 million square kilometers and South at 4 million. On a conformal map, North occupies 80 printed squares and South 60. On an equal-area map they occupy 40 and 80.

This packet uses fictional data to isolate a mechanism. It is not an empirical estimate.

Mechanism and worked example

A projection transforms a curved surface to a plane and makes tradeoffs among area, shape, direction and distance. Equal-area maps preserve relative areas, while conformal maps preserve local angles. Cartographers should select the projection for the task and state limitations. Choropleth population maps usually need rates or densities rather than raw totals.

Measured South/North area = 4/2 = 2. Equal-area printed ratio 80/40 = 2. Conformal printed ratio 60/80 = .75 would reverse the comparison if mistaken for ground area.

Complication

The conformal map is useful for preserving local angles; its printed areas are not ground areas. Neither flat map preserves every globe property. Density also needs measured population and area, not printed size.

Your investigation

Sketch the two equal-area rectangles at a shared scale, contrast the conformal counts, and write a caption identifying which map answers an area question.

Revise your answer to “Can a visually larger country have a smaller measured area?” using both source IDs. Separate the checked calculation from a judgment, and identify evidence that could reverse your judgment.

Vocabulary

Projection: A transformation representing the curved Earth on a flat surface.

Assumption: A condition accepted for a model or argument; it needs justification before applying the result elsewhere.

Independent extension

Same sphere, different preserved properties

Predict: at 60° north, does preserving local angles also preserve area? This original graticule represents a unit sphere, not country outlines or measured Earth data.

Prediction ProjectionSpherical MercatorLambert cylindrical equal-area, standard parallel 0° Cell center latitude0° north30° north60° north70° north

The outlined cell runs from 10° west to 10° east and from 55° to 65° latitude. Equal angular cells are not equal ground areas at different latitudes.

At the center, east–west length scale is 2.000000 and north–south length scale is 2.000000, relative to infinitesimal ground lengths. Their product is local area factor 4.000000.

The whole cell has spherical area 0.030423 radius-squared units and projected area 0.122948 matching square units. Its exact finite area ratio is 4.041268. This need not equal the center's local factor.

Formulas, units and primary methodology

Longitude λ and latitude φ enter in radians (degrees × π/180). Radius is 1, central longitude is 0, with no false offsets. Both have x = λ. Mercator y = asinh(tan φ), equivalently ln(tan(π/4 + φ/2)); equal-area y = sin φ. A cell's spherical area is Δλ × (sin φ_high − sin φ_low); its projected rectangle area is Δλ × (y_high − y_low).

Along a parallel, a tiny ground length is cos φ × Δλ; northward it is Δφ. Differentiating the projected coordinates gives east scale 1/cos φ in both, north scale 1/cos φ for Mercator and cos φ for equal-area. Equal directional scales preserve local angles; a product of 1 preserves area. These are spherical formulas, not ellipsoidal surveying.

Methodology: PROJ spherical Mercator formulas, PROJ equal-area standard parallels, and EPSG 9834 spherical equal-area formula transcription, citing Snyder, USGS Professional Paper 1395. No map imagery or country data are reproduced.

Transfer: switch to 70°, compare both projections, then test the equator. Explain why equal-area does not mean equal lengths in every direction. Would either cell's printed size alone justify a population-density comparison?

Revised explanation and transfer

Compare your reasoning

At 60°, Mercator stretches each local direction by 2, multiplying local area by 4. Equal-area stretches east–west by 2 and compresses north–south by 1/2, keeping area but changing shape. A finite Mercator cell spans latitudes with different scale, so its ratio is not exactly 4. Both have local unit scale at the equator, yet a finite Mercator cell still includes distorted latitudes. Density requires population and an appropriate ground-area denominator, not printed map area.

Copy your analysis report

{ "activity": "GEO-01.b spherical projection scales", "audit": { "projection": "mercator", "latitude": 60, "low": 55, "high": 65, "yLow": 1.1542345536088654, "yHigh": 1.506454237331945, "east": 1.9999999999999996, "north": 1.9999999999999996, "localArea": 3.9999999999999982, "ground": 0.030423093459356076, "printed": 0.12294786342601635, "finiteRatio": 4.0412676505191465 }, "worksheet": { "prediction": "", "revision": "" }, "feedbackSeen": false, "limits": "Original unit-sphere geometry, not observed country data or ellipsoidal Earth coordinates. Cell spans longitude -10 to 10 and latitude center ±5 degrees. Local scales are infinitesimal; finite-cell ratio is exact under this sphere. Temporary practice, no mastery assessment." }

Reset projection practice

Temporary worksheet: reset clears writing; guidance/report exposure remains during this page visit. Reload clears all work. This does not establish independent mastery.

Independently extend the investigation “Choose a projection for the question” with a documented real dataset or scholarly source. Preserve definitions, recreate one calculation or evidence table, compare a rival explanation and report why the teaching example cannot establish the real-world conclusion.