Golden Ratio Calculator

A golden ratio calculator splits a length into two parts whose proportion is φ = 1.6180339887, the golden section. Enter a 1,200 pixel width and the split is 741.64 and 458.36 pixels. The longer piece is always 61.803399% of the whole and the shorter piece 38.196601%, in any unit.

Three sister tools cover the neighboring questions. The Fibonacci Calculator generates the integer sequence whose consecutive ratios converge on φ, the Ratio Calculator simplifies and scales any ratio you already have, and the Aspect Ratio Calculator handles screen and image formats such as 16 : 9. Use this page when you have one length and want the two golden pieces, or a golden rectangle, in real units.

What the golden ratio actually is

Two positive lengths a (the longer part) and b (the shorter part) are in the golden ratio when the whole is to the longer part exactly as the longer part is to the shorter part:

Write x for the ratio a ÷ b. The definition becomes x = 1 + 1 ÷ x, which rearranges to the quadratic x² = x + 1. Its positive root is the constant everything else follows from:

Because x² = x + 1, the constant has two shortcuts that make hand calculation easy: φ² = φ + 1 = 2.6180339887 and 1 ÷ φ = φ − 1 = 0.6180339887. Those two identities are the reason the golden ratio keeps reappearing in geometry: it is the only positive number that is exactly one more than its own reciprocal.

The three ways to solve it

Depending on which measurement you already have, one of three formulas applies. This calculator switches between them with the choice cards at the top:

• You know the total (a + b): longer part a = total ÷ φ, shorter part b = total ÷ φ². Equivalently, a = 0.618034 × total and b = 0.381966 × total.
• You know the longer part (a): shorter part b = a ÷ φ, total = a × φ.
• You know the shorter part (b): longer part a = b × φ, total = b × φ².

All three are the same relationship read from a different starting point, so the answers always agree. A useful sanity check is that the longer part divided by the shorter part comes out as 1.618034 every time, whichever mode you used.

Worked example: a 1,200 pixel layout

Suppose you are laying out a content area 1,200 pixels wide and want a main column and a sidebar in golden proportion. Set the mode to Total and enter 1200.

• Longer part: 1,200 ÷ 1.6180339887 = 741.64 px (the main column).
• Shorter part: 1,200 ÷ 2.6180339887 = 458.36 px (the sidebar).
• Check the sum: 741.64 + 458.36 = 1,200.00 px, exactly the width you started with.
• Check the ratio: 741.64 ÷ 458.36 = 1.6180, which is φ to four decimals.

Rounding to whole pixels gives 742 and 458, whose ratio is 1.6201 instead of 1.6180. That rounding error of about 0.13% is invisible on screen, and it is the normal price of turning an irrational proportion into integers. If you would rather keep the split exact, divide the longer part again: 741.64 ÷ φ = 458.36, and 458.36 ÷ φ = 283.28. Those nested values are what the successive divisions table in the calculator lists for you.

Worked example: a 96 inch wall

The same arithmetic works in imperial units because φ is a pure number with no units attached. Say you are hanging a picture rail on a 96 inch (8 foot) wall and want the vertical divider placed at the golden section rather than the middle:

• Longer part: 96 ÷ 1.6180339887 = 59.33 in, which is 4 feet 11.33 inches.
• Shorter part: 96 ÷ 2.6180339887 = 36.67 in, which is 3 feet 0.67 inches.
• Sum check: 59.33 + 36.67 = 96.00 in.
• Ratio check: 59.33 ÷ 36.67 = 1.6180.

Compare that with a simple halving at 48 inches and 48 inches. The golden split moves the divider about 11.33 inches off center, which is exactly the point: it produces an obvious asymmetry that still feels ordered rather than accidental. Switch the unit selector to in or ft and the calculator labels the results in the unit you are actually measuring with.

Golden section of common lengths

The table below applies total ÷ φ and total ÷ φ² to lengths that come up often in design and building work. Every row splits into the same 61.803399% and 38.196601% shares, so you can scale any row up or down freely.

Notice that the values scale linearly. Doubling the total doubles both parts, which is why a single percentage pair, 61.803399% and 38.196601%, describes every row at once.

Golden rectangles

A golden rectangle has its long side and short side in the ratio φ : 1. Given a width, the golden height is width ÷ φ; given a height, the golden width is height × φ. The rectangle has a property no other shape shares: cut a square off its short side and the leftover rectangle is golden again, at 1 ÷ φ of the original size. Repeating that cut is how the familiar golden spiral diagram is drawn, one quarter circle per square.

A 16 by 10 print or screen is close to golden: the exact golden height for a 16 unit width is 9.89, so 16 : 10 is off by roughly 1.1%. The far more common 16 : 9 format has a ratio of 1.7778, which is about 9.9% wider than golden, so it is a different proportion rather than an approximation of one.

Fibonacci numbers as whole-number stand-ins

Consecutive Fibonacci numbers converge on φ when you divide one by the one before it. That gives you a ladder of integer ratios you can use when fractional pixels or awkward measurements are impractical. The error column below is the percentage by which each ratio misses 1.6180339887.

The ratios alternate above and below φ and the error falls by roughly a factor of 2.6 each step, so 13 : 8 is already within half a percent and 55 : 34 within a quarter of a tenth of a percent. To generate the underlying sequence to any length, use the Fibonacci Calculator.

Powers of phi

Because φ² = φ + 1, every power of φ is a whole number plus a multiple of φ, and the powers grow by a constant factor. These are handy when you build a proportional scale rather than a single split.

Starting from a 16 unit body size, a φ based type scale runs 16, 25.89, 41.89, 67.78 and 109.67, which in practice rounds to 16, 26, 42, 68 and 110. The steps are large, so many designers use a smaller ratio for text and reserve the φ jump for display sizes.

How to use this golden ratio calculator

1. Pick what you know. Choose Total if you have the full length, Longer if you have the bigger of the two parts, or Shorter if you have the smaller one.
2. Enter the measurement. Type any positive number. The quick buttons load common values such as 100, 1200 and 1920.
3. Choose the unit. Pixels, inches, feet, centimeters or millimeters. The unit is a label only, since the ratio itself is unitless.
4. Set the decimals. Zero decimals for whole pixels, two for print measurements, three when you are checking the arithmetic.
5. Read the results. The headline shows the longer and shorter parts side by side with the ratio between them, and the bar underneath draws the split to scale.
6. Scroll for the extras. Golden rectangle dimensions both ways, six successive divisions for nested layouts, and the Fibonacci table if you need integer stand-ins.

Everything recalculates as you type, so you can drag a value up and down and watch the split follow.

Who this calculator is for

• Web and product designers sizing a content column against a sidebar, or a hero against the section below it.
• Graphic designers and photographers placing a horizon, a fold or a focal point off center in a deliberate way.
• Woodworkers and furniture makers proportioning drawer fronts, shelf spacing or a box lid against its base.
• Architects, remodelers and home decorators dividing a wall, a window opening or a garden bed.
• Students checking homework on φ, golden rectangles and the Fibonacci connection.
• Anyone curious about why 1.618 shows up in so many places and what it actually means.

Key terms explained

• Phi (φ): the constant 1.6180339887498949, the positive root of x² = x + 1.
• Golden section: the point that divides a length into the golden ratio, at 61.803399% of the way along.
• Golden rectangle: a rectangle whose sides are in the ratio φ : 1.
• Golden spiral: the curve drawn through the squares you cut off successive golden rectangles; it grows by a factor of φ every quarter turn.
• Divine proportion: an older name for the same ratio, used in Renaissance mathematics texts.
• Reciprocal (1 ÷ φ): 0.6180339887, which equals φ − 1. Some sources call this number the golden ratio too.
• Irrational number: a number that cannot be written as a fraction of two whole numbers. φ is irrational, so every decimal you see is a rounded value.

What changes the result

The ratio itself never changes, so only three things move your numbers:

• The length you enter: results scale in direct proportion, so doubling the input doubles both parts.
• Which quantity you enter: entering 100 as the total gives 61.80 and 38.20, while entering 100 as the longer part gives 61.80 as the shorter part and 161.80 as the total. The mode selector matters more than any other setting.
• Rounding: displaying zero decimals can shift a split by up to half a unit and change the measured ratio slightly, as the 742 / 458 example shows.

The unit selector changes only the label. Ten inches and ten centimeters split into the same 6.18 and 3.82 in their own units.

Practical tips

• Remember 62 / 38. For a rough check without a calculator, the golden split is very close to 62% and 38% of the whole.
• Use the reciprocal for quick math. Multiplying by 0.618034 is the same as dividing by 1.618034 and is easier to do in your head.
• Round at the end, not in the middle. Rounding the longer part before you subtract it from the total introduces an error you then carry through the whole layout.
• Nest instead of guessing. To subdivide a golden layout, divide the longer part again rather than inventing a new proportion.
• Switch to Fibonacci when you need integers. A 13 : 8 or 21 : 13 split gives round numbers within half a percent of golden.
• Do not force it. Content, legibility and structural constraints beat proportion. Use φ as a starting point and adjust.

Limitations and assumptions

• The calculator works in double-precision floating point, which is accurate to roughly 15 significant digits. That is far more precision than any physical measurement needs, but φ itself has no exact decimal form.
• Results are rounded only for display. Internally the full precision constant is used, so a 0 decimal display can look one unit off from a hand calculation you rounded early.
• The tool takes a single length. It does not check that your two parts were already golden, and it does not adjust for margins, gutters, kerf or material thickness.
• It makes no aesthetic claim. The golden ratio is a proportional device, not a guarantee that a layout will look good.
• Popular claims about the ratio appearing in the Parthenon, in the Mona Lisa or in human body proportions are contested in the scholarly literature and are not asserted here.

How it compares to related calculators

This page answers "how do I split this length at the golden section?" If your question is a little different, a sister tool fits better:

• To build the integer sequence behind φ or find the nth term, use the Fibonacci Calculator.
• To simplify, scale or compare a ratio you already have, such as 742 : 458, use the Ratio Calculator.
• To resize an image or video to a fixed format such as 16 : 9, use the Aspect Ratio Calculator.
• To solve a proportion of the form a : b = c : x, use the Proportion Calculator.
• To turn the split into a share of a whole, use the Percentage Calculator.
• To evaluate √5 or other roots directly, use the Square Root Calculator; for powers of φ, the Exponent Calculator does the arithmetic.

Sources

Every number on this page is deterministic mathematics that follows from a single exact definition, so no external data source is involved:

• The constant: φ is defined as the positive root of x² = x + 1, that is (1 + √5) ÷ 2 = 1.6180339887498949. This is an exact definition, not a measured value.
• The split formulas: longer = total ÷ φ and shorter = total ÷ φ² follow algebraically from (a + b) ÷ a = a ÷ b.
• The Fibonacci table: generated from F(n) = F(n−1) + F(n−2) with F(0) = 0 and F(1) = 1, then divided pair by pair. Exact integer arithmetic, no source needed.
• Units: the inch and foot used in the imperial examples are defined exactly in terms of the meter (1 inch = 25.4 mm) by NIST Special Publication 811, Guide for the Use of the International System of Units. The ratio itself is unitless and unaffected.