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Learn scaling, one problem at a time

These six guides exist because the calculators kept answering questions correctly while people still got the wrong result — the direction was inverted, the percentage meant something else, one axis was ignored, or the print quietly rescaled itself. Each guide takes one of those failures, derives the mathematics behind it, and shows it in a figure drawn from the same numbers as the text. Start from the question that sounds like yours.
  • 6 guides
  • 7 purpose-built figures
  • Worked examples from the same engine

How the guides relate to the calculators

The split is deliberate. A calculator owns the numbers for your specific case: it parses your input, holds the factor as an exact fraction and reports the size, percentage, area and volume that follow from it. A guide owns the reasoning that is the same for everybody: why the formula is what it is, which readings are different answers rather than rounding differences, and how the result fails in practice.

So the guides never repeat a calculator's interface, and the calculators never try to teach the concept from scratch. If you want the shared definitions, the formulas and the rounding and provenance policy in one place instead, that is the methodology page. If you are looking for a documented convention rather than an explanation, use the common scales reference. If you already know what to compute, go straight to the workbench.