Golden Ratio Calculator

The golden ratio is a fixed proportion, usually written with the Greek letter phi (φ). Two positive lengths are in this proportion when the whole divided by the longer part equals the longer part divided by the shorter part. The common decimal approximation is 1.6180339887.

This calculator starts with the longer section, labelled a. It finds the matching shorter section b and the combined whole a + b. That starting point matters: entering the whole length as though it were a will produce a different division. The calculation is useful for geometric exercises, layout experiments, woodworking plans and checking a proposed proportion. It does not establish that a design is objectively attractive; aesthetic claims about φ are separate from its mathematical definition.

1. Measure or choose the longer section, a. It must be a positive length.
2. Select the unit that describes that length: millimetres, centimetres, metres, inches or feet.
3. Calculate. The page divides a by φ to obtain b, then adds a and b to obtain the whole.
4. Keep the displayed unit with both outputs. The proportion itself is unitless, so the same arithmetic works in any consistent length unit.

The calculator does not convert between units. If a is entered as 10 cm, both results are centimetres. Selecting metres changes the label and meaning, not the digits. For a drawing that must fit a specified total width, first derive the longer part as whole ÷ φ; this page instead expects the longer part itself.

For a physical project, decide which dimension is constrained before using the result. A frame may have a fixed outside width, while an image panel may start with a fixed longer side. Allow separately for borders, saw kerfs, seams and material thickness; those practical allowances are not part of the golden-ratio equation.

Results are rounded to four decimal places. Use more precision in an engineering or manufacturing plan if tolerances require it, and round only after choosing practical dimensions.

The two outputs specify a mathematically golden division based on the entered longer part. They can be transferred to a sketch or cut list after confirming scale and tolerances. A nearby practical measurement can still be perfectly suitable: 16.18 cm rather than 16.1803 cm may be more realistic for a hand tool, while digital fabrication may retain more digits.

Finding φ in a measured object requires attention to uncertainty. A ratio of 1.62 is close, but the conclusion depends on measurement accuracy and which edges were selected. The calculator proves only what follows from the supplied a; it does not test whether an existing rectangle or artwork was intentionally designed with the golden ratio.