Golden Ratio Calculator
Divide a length in the golden ratio phi = 1.6180339887, size a golden rectangle, or explore the golden angle and its link to Fibonacci numbers. Enter one value, choose the mode and what the value represents, and the calculator returns every related dimension, proportion and angle.
Formula
Worked example
Split a 100-unit length by phi: a = 100 / 1.6180 = 61.8034, b = 100 - 61.8034 = 38.1966. Check: a / b = 61.8034 / 38.1966 = 1.6180 = phi. Golden rectangle: width 100, length = 100 x 1.6180 = 161.8034, area = 16180.34.
What the golden ratio is
The golden ratio, written with the Greek letter phi, is the number where a whole length divided by its longer part equals that longer part divided by its shorter part. Both quotients equal phi = (1 + sqrt(5)) / 2 = 1.6180339887..., an irrational number with an infinite, non-repeating decimal expansion. It also satisfies the elegant equation phi^2 = phi + 1, and its reciprocal 1/phi = phi - 1 = 0.6180... Unlike most irrational numbers it pops up in geometry, plant growth patterns, and centuries of design practice.
How to split a length in the golden ratio
To divide a length so the two parts are in golden proportion, divide the whole by phi to get the longer segment, then subtract that from the whole to get the shorter one. If you know the longer segment, divide by phi to step down to the shorter part, or multiply by phi to step up to the whole. If you know the shorter segment, multiply by phi for the longer part, or by phi^2 for the whole. Each adjacent pair in the sequence (shorter, longer, whole) differs by exactly one factor of phi, mirroring how consecutive Fibonacci numbers approach the same ratio.
Golden rectangles and the golden spiral
A golden rectangle has a length-to-width ratio equal to phi. If you cut a square from one end, the leftover rectangle is itself golden, one scale smaller. You can keep cutting forever, each remnant a smaller golden rectangle. Tracing quarter-circle arcs through the corners of successive squares draws the golden spiral, a logarithmic spiral with a growth rate of phi per quarter turn. The same curve appears in nautilus shells, the seeds of sunflowers, and the arms of some galaxies, though these are approximations rather than exact golden spirals.
The golden angle and why plants use it
The golden angle is approximately 137.5077 degrees. It divides a full 360-degree rotation in the golden ratio (in the same way that a line splits a segment): the larger arc is 222.5 deg, the smaller is 137.5 deg, and their ratio is phi. Because phi is irrational, successive objects placed 137.5 deg apart never quite align, so every ray from the center receives roughly equal numbers of objects. Plants that arrange leaves, petals or seeds by the golden angle pack more efficiently than any other angle. The calculator always shows this constant so you can cross-check design layouts.
The Fibonacci sequence and convergence to phi
The Fibonacci sequence starts 1, 1, 2, 3, 5, 8, 13, 21... where each number is the sum of the previous two. The ratio of consecutive terms, F(n)/F(n-1), oscillates above and below phi and converges to it. By F(15)/F(14) = 610/377 = 1.6180327..., the approximation is accurate to 7 decimal places. Switch the calculator to Fibonacci mode to see a live table and chart of this convergence, with the exact difference from phi at each step.
Golden-ratio splits of common lengths
| Whole (a + b) | Longer segment a | Shorter segment b | a as % of whole |
|---|---|---|---|
| 1 | 0.6180 | 0.3820 | 61.80% |
| 10 | 6.1803 | 3.8197 | 61.80% |
| 16 | 9.8885 | 6.1115 | 61.80% |
| 100 | 61.8034 | 38.1966 | 61.80% |
| 360 | 222.4922 | 137.5078 | 61.80% |
| 500 | 309.0170 | 190.9830 | 61.80% |
| 1000 | 618.0340 | 381.9660 | 61.80% |
| 1920 | 1186.625 | 733.375 | 61.80% |
Longer and shorter segments when the whole length is divided by phi = 1.6180339887.
Frequently asked questions
What is the exact value of the golden ratio?
The golden ratio is phi = (1 + sqrt(5)) / 2, which is approximately 1.6180339887498948482... It is irrational, so its decimal expansion goes on forever without repeating. Any value you see is rounded; this calculator lets you choose 2, 4, 6 or 8 decimal places to control how many digits appear.
How is the golden ratio related to the Fibonacci sequence?
As you go further along the Fibonacci sequence (1, 1, 2, 3, 5, 8, 13, 21...), the ratio of each number to the previous one gets closer and closer to phi. For example 21/13 = 1.6154, 34/21 = 1.6190, and 610/377 = 1.61803..., already matching phi to 7 decimal places. Switch the calculator to Fibonacci mode to see the full convergence table.
Why does the longer segment always come out to about 61.8% of the whole?
Dividing the whole by phi gives the longer segment, and 1/phi = 0.6180... So the longer part is always approximately 61.80% of the total and the shorter part is the remaining 38.20%, regardless of the length you start with.
What is a golden rectangle?
A golden rectangle is any rectangle whose length-to-width ratio equals phi (approximately 1.618). It has the unique property that cutting a square off one end leaves a smaller rectangle that is also golden. Use Rectangle mode in this calculator to find the missing side given one known side.
What is the golden angle and where does it come from?
The golden angle is approximately 137.5077 degrees. It is the smaller of the two angles produced when a full 360-degree turn is divided in the golden ratio: 360 / phi^2 = 137.508 deg. Plants use it to arrange leaves and petals because the golden angle is irrational, meaning successive items placed at that angle never exactly repeat, so each gets maximum exposure to light and space.
Can I use the golden ratio in typography and screen design?
Yes. A common approach is to pick a body font size and multiply it by phi to get a heading size, or to divide a column width by phi to find a sidebar width. For example, a 1920 px canvas divided by phi gives a main area of about 1187 px and a sidebar of about 733 px. Enter 1920 in Segment mode with the "whole length" role to get those numbers instantly.