Phi Symbol (Φ φ): Meaning, Golden Ratio, and How to Type It

Phi is the twenty-first letter of the Greek alphabet and one of the most heavily used symbols in science. The lowercase φ is the golden ratio φ ≈ 1.618, the azimuthal angle in spherical coordinates, a phase angle, the standard normal density, and Euler's totient function. The uppercase Φ stands for magnetic and electric flux, the normal cumulative distribution function, electric potential, and the work function of a metal. Lowercase phi also comes in two shapes, the loopy φ and the straight ϕ, which LaTeX calls \varphi and \phi. To type phi, use \phi, \varphi, or \Phi in LaTeX, the HTML entity φ, or the F key on a Greek keyboard layout.

This guide covers what each form of phi means, where it appears in math, physics, statistics, and engineering, and how to type φ, ϕ, and Φ on Mac, Windows, iPhone, in HTML, and in LaTeX.

Phi Symbol Copy and Paste

Copy all three forms from the box below (use the copy button), or take the code you need from the table:

1Φ φ ϕ
Copy

Watch the LaTeX names: \phi is the straight ϕ and \varphi is the loopy φ, which is the reverse of what many people expect. Most fonts draw U+03C6 as the loopy φ, which is why \varphi is the closer match to what you see in a browser.

The Phi Symbol Explained

Phi comes in three shapes, all of which you can copy from here:

• Φ — uppercase phi (Unicode U+03A6)
• φ — lowercase phi, the loopy or "curly" form (Unicode U+03C6)
• ϕ — straight or "stroked" phi, a circle with a vertical bar (Unicode U+03D5)

The name is pronounced "fie" (rhyming with "sky") by most English-speaking mathematicians and "fee" in modern Greek and in many classrooms. In the Greek numeral system phi stood for 500. Phi represents the "ph" sound of philosophy and physics, and it sits on the F key of Greek keyboards.

φ vs. ϕ: \varphi and \phi

The two lowercase shapes have no difference in meaning, only in typography, but the names are confusing. In LaTeX, \phi produces the straight ϕ and \varphi produces the loopy φ. In Unicode, however, the plain code point U+03C6 "Greek small letter phi" is the loopy φ, and U+03D5 "Greek phi symbol" is the straight ϕ. So the default LaTeX phi is the Unicode variant, and vice versa. In Greek text and handwriting the loopy form is standard; in printed mathematics both are common, and some authors use both in one paper for two different quantities.

Uppercase Phi (Φ)

Uppercase phi does not resemble any Latin letter, so it is used freely as a symbol, chiefly for fluxes and potentials.

Uses of uppercase Φ:

• Magnetic flux — Φ_B, measured in webers
• Electric flux — Φ_E in Gauss's law
• Normal CDF — Φ(x), the cumulative distribution function of the standard normal
• Electric potential — Φ, with E = −∇Φ
• Gravitational potential — Φ in Newtonian gravity and cosmology
• Work function — Φ in the photoelectric effect

Magnetic Flux and Faraday's Law

Magnetic flux is the amount of magnetic field passing through a surface:

Φ_B = ∫ B · dA

Its unit is the weber (1 Wb = 1 T·m²). Faraday's law of induction says that a changing flux through a loop induces an electromotive force:

ε = −dΦ_B / dt

which is the principle behind generators, transformers, and induction cooktops. The electric flux Φ_E = ∫ E · dA plays the same role in Gauss's law, Φ_E = Q_enclosed / ε₀.

Φ(x): The Normal Distribution

In statistics Φ(x) is the cumulative distribution function of the standard normal distribution, and φ(x) is its density:

φ(x) = (1/√(2π)) e^(−x²/2), Φ(x) = ∫_{−∞}^{x} φ(t) dt

Φ has no closed form, which is why z-tables exist: Φ(1.96) ≈ 0.975 is the number behind the 95% confidence interval, and Φ(0) = 0.5.

Work Function and Potentials

In the photoelectric effect, Φ is the work function, the minimum energy needed to free an electron from a metal's surface. Einstein's 1905 equation gives the maximum kinetic energy of the ejected electron:

K_max = hν − Φ

with values of a few electron-volts (about 4.5 eV for copper). Φ (or lowercase φ) is also the standard symbol for the electrostatic potential, E = −∇Φ, and for the gravitational potential, g = −∇Φ.

Lowercase Phi (φ, ϕ)

Uses of lowercase φ:

• Golden ratio — φ = (1 + √5)/2 ≈ 1.618
• Azimuthal angle — φ in spherical and cylindrical coordinates
• Phase — φ in A cos(ωt + φ) and in complex numbers z = re^(iφ)
• Euler's totient — φ(n) in number theory
• Normal density — φ(x) in statistics
• Phi coefficient — φ, the correlation for two binary variables

The Golden Ratio

The golden ratio is the positive solution of x² = x + 1:

φ = (1 + √5) / 2 ≈ 1.6180339887

Two quantities are in the golden ratio when the ratio of their sum to the larger equals the ratio of the larger to the smaller. The letter was chosen in the early twentieth century by the mathematician Mark Barr after the sculptor Phidias, who is said to have used the proportion in the Parthenon. The ratio has unusual algebraic properties:

• φ² = φ + 1 ≈ 2.618
• 1/φ = φ − 1 ≈ 0.618
• φ = 1 + 1/(1 + 1/(1 + …)), the simplest continued fraction

The ratio of consecutive Fibonacci numbers converges to φ: 8/5 = 1.6, 13/8 = 1.625, 21/13 ≈ 1.615, 34/21 ≈ 1.619. It shows up in the regular pentagon, the logarithmic spiral, and the arrangement of sunflower seeds. Claims that it governs the human body or every great painting are mostly myth.

Angles and Phase

In spherical coordinates physics texts write a point as (r, θ, φ), where φ is the azimuthal angle in the xy-plane measured from the x-axis and θ is the polar angle from the z-axis. Many mathematics textbooks swap the two, writing (ρ, θ, φ) with θ as the azimuth, so check the convention.

Phi is also the default symbol for a phase angle. A sinusoidal signal is written A cos(ωt + φ), a complex number in polar form is z = r e^(iφ) with φ = arg z, and in AC circuits the power factor is cos φ, the cosine of the phase difference between voltage and current.

Euler's Totient Function

In number theory φ(n) counts the positive integers up to n that are coprime to n. For a prime p, φ(p) = p − 1; for n = 12 the coprime integers are 1, 5, 7, and 11, so φ(12) = 4. In general, if n = p₁^a₁ p₂^a₂ ⋯ then:

φ(n) = n ∏ (1 − 1/pᵢ)

The function is multiplicative and underlies Euler's theorem, a^φ(n) ≡ 1 (mod n) when gcd(a, n) = 1, which generalizes Fermat's little theorem and is the mathematical basis of the RSA encryption algorithm, where the private key is computed modulo φ(n) = (p − 1)(q − 1).

Phi in Different Fields

Mathematics

• The golden ratio φ ≈ 1.618
• Euler's totient function φ(n)
• Argument of a complex number, φ = arg z
• Azimuthal angle in spherical and cylindrical coordinates

Physics

• Magnetic flux Φ_B and electric flux Φ_E
• Electric and gravitational potentials Φ
• Work function Φ in the photoelectric effect
• Scalar fields φ in quantum field theory, including the Higgs field

Engineering

• Phase angle φ and power factor cos φ in AC circuits
• Porosity φ of rocks and soils, the fraction of pore volume

Statistics

• Standard normal density φ(x) and CDF Φ(x)
• Phi coefficient φ, the Pearson correlation between two binary variables, equal to √(χ²/n) for a 2 × 2 table

How to Type Phi

Windows: Phi is one of the few Greek letters with a classic Alt code: hold Alt and type 237 on the numeric keypad for φ or 232 for Φ, which works in almost every program. In Microsoft Word, Alt + 966 also works, and the most reliable method is to type 03C6 (or 03D5 for ϕ, 03A6 for Φ) and press Alt + X. You can also use Character Map or the emoji panel (Win + .) and its Symbols tab.

Mac: macOS has no Option-key shortcut for phi. Add the Greek keyboard in System Settings > Keyboard > Input Sources, switch to it, and press F for φ or Shift + F for Φ. Alternatively, open the Character Viewer with Control + Command + Space and search for "phi."

iPhone and iPad: Go to Settings > General > Keyboard > Keyboards > Add New Keyboard and choose Greek. Switch keyboards with the globe key; φ sits on the F key. Android users can add Greek in Gboard's language settings.

Word and Google Docs: In Word, use Insert > Symbol > More Symbols with the subset set to Greek and Coptic. In Google Docs, use Insert > Special characters and search for "phi"; the straight form is listed as "Greek phi symbol."

HTML: Write φ for φ, Φ for Φ, and ϕ (or ϕ) for ϕ.

LaTeX: In math mode, \phi gives the straight ϕ, \varphi gives the loopy φ, and \Phi gives Φ. All three are built in. The following document compiles as written:

1\documentclass{article}
2\usepackage{amsmath}
3\begin{document}
4Straight phi is $\phi$, loopy phi is $\varphi$, and capital phi is $\Phi$.
5\[
6  \varphi = \frac{1 + \sqrt{5}}{2} \approx 1.618,
7  \qquad
8  \Phi_B = \int \mathbf{B} \cdot d\mathbf{A},
9  \qquad
10  \Phi(x) = \int_{-\infty}^{x} \frac{1}{\sqrt{2\pi}} e^{-t^2/2} \, dt
11\]
12\[
13  K_{\max} = h\nu - \Phi,
14  \qquad
15  \phi(n) = n \prod_{p \mid n} \left(1 - \frac{1}{p}\right),
16  \qquad
17  z = r e^{i\phi}
18\]
19\end{document}
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For phi in running text, the textgreek package provides \textphi and \textPhi, which give upright letters that match the surrounding font.

Related Greek Letters

Phi is almost never alone: the azimuth φ pairs with the polar angle θ, the normal CDF Φ pairs with the standard deviation σ, the photoelectric equation pairs Φ with frequency ν, and the phase φ sits inside ωt + φ.

• Theta symbol (Θ θ) — the polar angle that accompanies φ in spherical coordinates
• Pi symbol (Π π) — the √(2π) in the normal density and the 2π in a full turn of φ
• Sigma symbol (Σ σ) — the standard deviation that scales the normal distribution
• Omega symbol (Ω ω) — angular frequency in cos(ωt + φ)
• Psi symbol (Ψ ψ) — the usual partner of φ for a second function or quantum state

The full list of letters is on our Greek alphabet reference, and Greek letters in math and science lists every letter's standard meanings. For a copy-and-paste palette, see the Greek symbol tool.

Related symbols and guides

• Phi symbol page — copy the symbol and its LaTeX command
• Greek letters reference — every symbol in the category with its LaTeX command
• Greek alphabet in LaTeX — how to typeset these symbols in a document
• Theta symbol (θ)
• Psi symbol (ψ)

FAQ

What does the phi symbol mean?

It depends on the field. In mathematics φ is usually the golden ratio 1.618 or Euler's totient function; in physics φ is an angle or phase and Φ is a flux or potential; in statistics Φ(x) is the standard normal CDF. The surrounding equation tells you which.

What is the golden ratio φ?

φ = (1 + √5)/2 ≈ 1.618, the number whose square is itself plus one. It is the limit of the ratio of consecutive Fibonacci numbers and appears in the geometry of the regular pentagon. It is sometimes written Φ or τ.

What is the difference between φ and ϕ?

They are two typographic forms of the same lowercase letter. The loopy φ (LaTeX \varphi, Unicode U+03C6) is standard in Greek text and handwriting; the straight ϕ (LaTeX \phi, Unicode U+03D5) is common in printed mathematics. Some authors use both to mean two different quantities, so check the paper's notation.

How do you pronounce phi?

Most English-speaking mathematicians say "fie" (rhyming with "pi"), while modern Greek speakers and many science teachers say "fee." Both are accepted.

What is the phi symbol?

The phi symbol is the Greek letter phi: Φ in uppercase and φ or ϕ in lowercase. It stands for the golden ratio 1.618, Euler's totient function, magnetic and electric flux, the work function, the normal CDF Φ(x), angles and phase, depending on the field.

How do you type the phi symbol?

On Windows hold Alt and type 237 on the numeric keypad for φ or 232 for Φ. On a Mac add the Greek input source and press F for φ or Shift + F for Φ. In HTML write φ or Φ, or copy Φ φ ϕ from the box at the top of this page.

Should I use \phi or \varphi in LaTeX?

Either is correct; they are the same letter in two shapes. \varphi (loopy φ) is the common choice for the golden ratio and in most modern texts, while \phi (straight ϕ) is common in older physics books. Pick one and use it consistently, and use \Phi for the uppercase letter.

Writing with Greek Symbols

Papers that mix golden ratios, fluxes, and phase angles need phi typeset cleanly and consistently, with a deliberate choice between \phi and \varphi. In LaTeX that means the command in math mode rather than a pasted Unicode character, which keeps spacing and italics correct throughout the document.

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