Golden Rectangle Calculator
Enter any one dimension of a golden rectangle and get all the others instantly. Provide the long side, short side, diagonal, area, or perimeter and the calculator returns the complete set of measurements using the golden ratio phi (approximately 1.618). Switch between metric and imperial units and see the step-by-step working for each result.
Formula
Worked example
Given a long side of 100 cm: b = 100 / 1.6180339887 = 61.8034 cm. Diagonal = sqrt(100^2 + 61.8034^2) = sqrt(10000 + 3819.66) = sqrt(13819.66) = 117.557 cm. Area = 100 * 61.8034 = 6180.34 cm^2. Perimeter = 2*(100 + 61.8034) = 323.607 cm.
What is a golden rectangle?
A golden rectangle is a rectangle whose sides are in the golden ratio, phi (approximately 1.618033988...). That means the long side divided by the short side equals the same constant you get when you divide the sum of both sides by the long side alone. The golden ratio is the only positive number with the property phi^2 = phi + 1 and 1/phi = phi - 1, which is why it appears in so many natural and designed objects. Remove a square from a golden rectangle and the piece that remains is itself a golden rectangle, rotated 90 degrees. Repeat the process infinitely and you trace out the golden spiral, the curve commonly associated with the nautilus shell, sunflower seed arrangements, and the spiral arms of some galaxies.
How this calculator works
Choose which dimension you already know from the dropdown: long side (a), short side (b), diagonal (d), area (A), or perimeter (P). Enter its value and every other dimension is computed in real time. Internally the calculator first derives the long side from your input using the relevant formula below, then computes every other quantity from a and phi. The ratio output confirms that a/b is indeed phi, which is useful when you want to verify an existing shape or check a design draft. Switch between metric (centimetres) and imperial (inches) by changing the Units selector; all labels update but the underlying ratio is dimensionless, so only the unit label changes.
Golden rectangle formulas
All formulas derive from the single constraint a/b = phi. From long side: b = a / phi. From short side: a = b * phi. From diagonal: d = a * sqrt(1 + 1/phi^2), so a = d / sqrt(1 + 1/phi^2). From area: A = a * b = a * (a/phi) = a^2/phi, so a = sqrt(A * phi). From perimeter: P = 2(a + b) = 2a(1 + 1/phi), so a = P / (2(1 + 1/phi)). In every case b = a / phi, area = a * b, diagonal = sqrt(a^2 + b^2), and perimeter = 2(a + b).
Real-world applications of the golden rectangle
Architects and product designers use the golden rectangle because the ratio is visually pleasing to most observers across cultures. Classic examples cited in design literature include the proportions of the Parthenon facade, credit cards (approximately 85.6 mm by 53.98 mm, giving a ratio of about 1.586), standard business cards, and many smartphone screen aspect ratios. In graphic design, laying out content within nested golden rectangles produces a grid called the Fibonacci grid, which many publications use for column widths. Fine artists use the golden section to position focal points on a canvas. Carpenters and woodworkers apply it to decide the proportions of furniture pieces such as tabletops and cabinet doors.
Golden ratio key identities
| Identity | Value | Meaning |
|---|---|---|
| phi = (1 + sqrt(5)) / 2 | 1.6180339887... | The golden ratio |
| 1 / phi | 0.6180339887... | Reciprocal equals phi minus 1 |
| phi^2 | 2.6180339887... | Square equals phi plus 1 |
| phi - 1 | 0.6180339887... | 1 / phi |
| a / b | phi | Long to short side ratio |
| b / a | 1 / phi | Short to long side ratio |
Exact algebraic identities that define phi and its reciprocal.
Frequently asked questions
What is the golden ratio and where does it come from?
The golden ratio, written as phi, equals (1 + sqrt(5)) / 2, approximately 1.6180339887. It is the unique positive number satisfying phi^2 = phi + 1. It appears naturally when you divide a line segment so that the ratio of the whole to the longer part equals the ratio of the longer part to the shorter part. It also emerges as the limiting ratio of consecutive Fibonacci numbers (1, 1, 2, 3, 5, 8, 13, ...) as the sequence grows.
How do I calculate the golden rectangle dimensions from the short side?
Multiply the short side by phi (approximately 1.618034). For example, if the short side is 60 cm, the long side is 60 * 1.618034 = 97.082 cm. Select "Short side (b)" in the Known value dropdown and enter 60 to get all dimensions at once.
Can I calculate a golden rectangle from the diagonal?
Yes. The diagonal relates to the long side by d = a * sqrt(1 + 1/phi^2). Rearranging gives a = d / sqrt(1 + 1/phi^2), which is approximately a = d / 1.17557. Select "Diagonal (d)" from the dropdown and enter the diagonal length.
Is the golden rectangle the same as the golden ratio?
The golden ratio (phi) is the numeric constant 1.618...; the golden rectangle is a geometric shape whose side lengths are in that ratio. The rectangle is the most common way the ratio is expressed in 2D design and architecture, but the ratio itself appears in many other geometric contexts, including pentagons, Fibonacci spirals, and continued fractions.
What happens when you cut a square from a golden rectangle?
If you cut a square with side equal to the short side (b) from a golden rectangle, the remaining piece has dimensions b by (a - b). Because a = phi * b, we have a - b = (phi - 1) * b = (1/phi) * b. So the remaining rectangle has a long side of b and a short side of b/phi, which is again a golden rectangle. This self-similar property repeats indefinitely and generates the golden spiral when you draw quarter-circle arcs inside each successive square.
How do I find the area of a golden rectangle?
The area is simply the long side multiplied by the short side: A = a * b. Because b = a/phi, you can also write A = a^2 / phi. If you know only the area and want all other dimensions, select "Area (A)" from the dropdown, enter the area value, and the calculator will derive a = sqrt(A * phi) and then find all remaining dimensions.