How does a map projection change distances, angles and areas?

A map projection converts locations on a curved reference surface into planar coordinates. This is not merely a change of drawing size: different spatial properties can be altered in different ways. Reading a historical map therefore includes understanding its projection rather than treating every visible distance, angle and area as equally faithful.

Consider an equatorial equirectangular projection on a sphere of radius R. Longitude λ and latitude φ are measured in radians, and the map uses x = Rλ and y = Rφ. Equal angular increments receive equal spacing on the page. The equator is the standard parallel, where the local scale is true in both directions.

On the sphere, a small east–west distance at latitude φ is R cos φ times its longitude increment. The corresponding map distance lacks that cosine factor. The east–west scale is therefore 1/cos φ, while north–south scale remains 1 in this model.

The local area factor is the product of those scales. At 60°, it is 2: a tiny ground circle maps to an ellipse twice as wide but equally tall. At 45°, the factor is about 1.41; at the equator it is 1. These are local limits, not exact area ratios for large countries crossing many latitudes.

Other projections distribute distortion differently. The projection name, reference surface and scale conventions matter before map shapes become evidence for a geographical comparison. Enlarged high-latitude shapes do not alone show that those places occupy equally enlarged areas on Earth.