Proportion Scale Calculator

Proportional scaling is a similarity transformation: one factor k multiplies both dimensions, so the width-to-height ratio is preserved and the shape is unchanged. This tool accepts that factor three ways, as a percentage, a target width, or a target height, the last two solving for k first as target width divided by original width. A 1200 by 800 source at 50 percent becomes 600 by 400 and stays at 3:2; setting the width alone to 600 while leaving height at 800 would silently change the ratio to 3:4 and distort the content. The reduced ratio shown is the original pair divided by their greatest common divisor, which is why 1200 by 800 displays as 3:2. The area behaviour surprises people most often. Because area is a product of two scaled lengths, it changes with k squared, not k. Scaling to 200 percent quadruples area to 400 percent; scaling to 50 percent leaves 25 percent; a 1:100 architectural model has one ten-thousandth of the original plan area, which is why material and ink estimates cannot be scaled linearly. In three dimensions the same argument gives k cubed for volume and mass. Common target ratios include 3:2 for most DSLR frames, 4:3 for Micro Four Thirds, 16:9 for HDTV, and 1:sqrt(2), about 1.414, for ISO A-series paper. Two limits: the reduced ratio rounds inputs to integers before taking the common divisor, so decimal pairs display approximately, and the model is purely geometric, so enlarging a raster beyond its native pixel count produces soft output.

The mathematics is Greek and predates any design use. Book V of Euclid's Elements, compiled around 300 BC, presents the theory of proportion generally credited to Eudoxus of Cnidus, and Book VI applies it to similar figures, establishing that corresponding sides stand in fixed ratio and that areas stand in the duplicate ratio, which is the square law used here. Archimedes extended the reasoning to volumes in the third century BC. The result was applied practically long before it was written down, since Egyptian and Mesopotamian builders used squared grids to transfer figures between sizes. Standardised scale ratios arrived with industrial drawing. Gaspard Monge's descriptive geometry, published in 1799, gave engineering drawing its formal basis, and the nineteenth century brought engraved scale rules carrying fixed reduction ratios. Metric practice settled on a preferred series, now codified in ISO 5455, specifying reductions such as 1:2, 1:5, 1:10 and their decimal multiples, giving the familiar 1:50 for detail plans, 1:100 for floor plans and 1:500 for site layouts. Model conventions grew separately from hobby and military modelling, producing 1:24, 1:48 and 1:72. The square-cube relationship acquired a famous biological reading through Galileo Galilei, who argued in Discorsi e dimostrazioni matematiche of 1638 that an animal cannot simply be scaled up because weight grows with the cube of length while bone cross section grows only with the square, an argument elaborated by D'Arcy Wentworth Thompson in On Growth and Form in 1917.