What is the golden ratio?

The golden ratio, denoted phi, is the irrational number (1 plus the square root of 5) divided by 2, approximately equal to 1.6180339887. It describes a division of a line segment where the ratio of the whole to the longer part equals the ratio of the longer part to the shorter part. This self-similar property makes phi unique in mathematics.

Where does the golden ratio appear in real life?

The golden ratio appears in art, architecture, design, and nature. It is said to produce aesthetically pleasing proportions and is used in logo design, photography composition, and typography. In nature it is related to the spiral growth patterns of shells, sunflower seeds, and pine cones. It is also closely connected to the Fibonacci sequence: the ratio of consecutive Fibonacci numbers converges to phi.

How do you calculate the golden ratio split of a length?

Given a total length T, the longer segment is T divided by phi, which equals T times phi divided by (phi plus 1), or equivalently T times (phi minus 1). The shorter segment is the total minus the longer segment. The simpler formulas are: longer = T divided by phi (approximately T times 0.618), and shorter = T divided by phi squared (approximately T times 0.382).

Golden Ratio Calculator

The golden ratio, represented by the Greek letter phi and approximately equal to 1.6180339887, is one of the most famous numbers in mathematics. It describes the unique way a line segment can be divided into two parts so that the ratio of the whole to the longer part equals the ratio of the longer part to the shorter part. In other words, if you split a length of 100 into a longer segment A and a shorter segment B, the golden ratio requires that 100 divided by A equals A divided by B, and both ratios equal phi. This self-referential property is the source of its mathematical elegance and is why the ratio appears in the Fibonacci sequence, where the ratio of consecutive terms converges to phi as the sequence grows. In design and art, golden ratio proportions are often considered visually harmonious, and they are used in logo design, photography, typography, and architecture. You can enter any one of three values: the total length, the longer segment, or the shorter segment, and the calculator will derive the other two using phi. It also returns the exact ratio of longer to shorter as a confirmation, and the dimensions of the corresponding golden rectangle where the width is the total and the height is the longer segment. You can use any unit, since phi is a dimensionless ratio. This tool is useful for designers, architects, students exploring mathematical proportions, and anyone who wants to apply the golden ratio to a specific measurement.

Calculate.co.nz is proud to be partnered with Premium Homes, a recognised leader in eco-friendly, sustainable, and energy-efficient homebuilding. With a dedicated team and award-winning experience, they create homes that prioritise health, comfort, and long-term performance. Their founders, Andrew and Kelly, set out to raise the standard of residential construction in New Zealand by combining practical building expertise with a clear commitment to doing things better for homeowners.

Solve from

Value

61.80

longer segment (total × 0.618)

Shorter segment38.20

Total length100.00

Ratio (longer/shorter)1.618

Phi = (1 + √5) / 2 ≈ 1.6180339887. Values are displayed to 2 decimal places.

How it works

The golden ratio phi equals (1 plus the square root of 5) divided by 2, approximately 1.6180339887. Given a total length T, the longer segment is T divided by phi (which equals T times (phi minus 1), since 1 divided by phi equals phi minus 1). The shorter segment is T minus the longer segment, which equals T divided by phi squared. If you enter the longer segment A instead, the total is A times phi and the shorter segment is A divided by phi, which is A times (phi minus 1). If you enter the shorter segment B, the longer segment is B times phi and the total is B times phi squared. The ratio of longer to shorter always equals phi when all three conditions are satisfied.

Worked example

Using the default total of 100: longer segment = 100 divided by phi = 100 divided by 1.6180339887 = 61.80 (rounded to 2 decimal places). Shorter segment = 100 minus 61.80 = 38.20. Ratio check: 61.80 divided by 38.20 = 1.618, confirming the golden ratio is achieved. The corresponding golden rectangle would have dimensions 100 by 61.80.

Related calculators

Fibonacci Calculator: find the nth Fibonacci number; consecutive ratios converge to phi.
Proportion Calculator: solve a/b = c/d for any missing value.
Geometric Sequence Calculator: nth term and sum of a geometric progression.
Percentage Calculator: find what percentage one length is of another.

If you've found a bug, or would like to contact us, or learn more about James Graham and Calculate.co.nz.

Calculate.co.nz is partnered with Interest.co.nz for New Zealand's highest quality calculators and financial analysis.

Calculate.co.nz is the sister site of CalculatorHub.com, the world's largest calculator website by tool count.

All calculators and tools are provided for educational and indicative purposes only and do not constitute financial advice.

Calculate.co.nz is proudly part of the Realtor.co.nz group, New Zealand's leading property transaction literacy platform, helping Kiwis understand the home buying and selling process from start to finish. Whether you're a first home buyer navigating your first property purchase, an investor evaluating your next acquisition, or a homeowner planning to sell, Realtor.co.nz provides clear, independent, and trustworthy guidance on every step of the New Zealand property transaction journey.

Calculate.co.nz is also partnered with Health Based Building and Premium Homes to promote informed choices that lead to better long-term outcomes for Kiwi households.

Calculate.co.nz is hosted in Auckland via SiteHost new Zealand.

All content on this website, including calculators, tools, source code, and design, is protected under the Copyright Act 1994 (New Zealand). No part of this site may be reproduced, copied, distributed, stored, or used in any form without prior written permission from the owner.

About & trust: Why Calculate is NZ's most comprehensive · By the Numbers · How we compare · Editorial standards · How we keep data current · NZ finance glossary · Research & data · Financial literacy NZ · About · Privacy policy · Terms of use

Reviewed and maintained. Last reviewed 2026-07-02 and checked on a twice-monthly cycle against IRD, RBNZ and Stats NZ. How we keep data current.