Which projects distorts shapes the most?

The correct answer is Lambert Conformal Conic

Conical projections are the most suitable cartographic projections for the representation of countries in medium latitudes and / or for the representation of a particular hemisphere. They are thought of as if a cone were placed over one of the terrestrial hemispheres. This cone being in contact with the Earth at mid-latitudes and the rest of the planet being projected towards the rest of the cone, then the cone is cut vertically and flattened.

On the cone's contact surface with the Earth, we can classify the conical projections as tangent or secant. The tangents are those where the middle latitudes only touch the cone, without passing it, while the secant passes through the lines of the cone. The tangent conical projection is ideal for representing some area at medium latitude, as it avoids distortions in this area. The secant conical projection, in turn, is ideal for representing a wider area, such as a hemisphere, as the distortion is smaller and better distributed among the various latitudes represented on the map.

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