Kappa Symbol (Κ κ): Meaning, Uses, and How to Type It

Kappa is the tenth letter of the Greek alphabet. The lowercase κ is the symbol for curvature in geometry, Cohen's kappa in statistics, thermal conductivity and the dielectric constant in physics, and the condition number in numerical analysis. The uppercase Κ looks exactly like the Latin K and is almost never used as a symbol, though the word "Kappa" lives on in fraternity names and as the best-known emote on Twitch. To type kappa, use \kappa in LaTeX (or \varkappa for the cursive ϰ), the HTML entity κ, or the K key on a Greek keyboard layout.

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

Kappa Greek Symbol: Copy and Paste

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

1κ Κ ϰ
Copy

In LaTeX, \kappa works in any math mode with no package, while \varkappa needs \usepackage{amssymb}. Kappa has no classic Windows Alt code: the decimal codes above work in Microsoft Word but not in every program, where Character Map or the emoji panel (Win + .) is the reliable way.

The Kappa Symbol Explained

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

• Κ — uppercase kappa (Unicode U+039A)
• κ — lowercase kappa (Unicode U+03BA)
• ϰ — "kappa symbol," a cursive variant with a rounded stroke (Unicode U+03F0), written \varkappa in LaTeX

The name is pronounced "KAP-uh" in English and "KAH-pa" in modern Greek. In the Greek numeral system kappa stood for 20. The letter descends from the Phoenician kaph and is the ancestor of the Latin K, which is why it sits on the K key of every Greek keyboard. Kappa represents a hard "k" sound, so Greek-derived words are usually spelled with c or k in English: kinetic, kilo, and cosmos all begin with kappa in Greek.

Uppercase Kappa (Κ)

Because uppercase kappa is visually identical to the Latin capital K, it has essentially no use as a mathematical symbol; writing Κ for a variable would simply be read as K. The capital letter appears in Greek words and in names:

• Fraternities and societies — Kappa Alpha, Kappa Kappa Gamma, Kappa Sigma, and the honor society Phi Beta Kappa, the oldest in the United States (founded 1776)
• Kappa on Twitch — the "Kappa" emote, a grayscale face of former Justin.tv employee Josh DeSeno added in 2009, used to mark sarcasm; typing "Kappa" in chat means "I'm joking"

None of these have anything to do with the mathematical κ, but they dominate search results, so it is worth knowing the difference. The Japanese kappa, a river creature of folklore, is an unrelated word that merely sounds the same.

LaTeX reflects the capital letter's redundancy: there is no \Kappa command. If you need an upright capital kappa in an equation, type \mathrm{K}.

Lowercase Kappa (κ)

Uses of lowercase κ:

• Curvature — κ of a curve, with radius of curvature R = 1/κ
• Cohen's kappa — inter-rater agreement in statistics
• Thermal conductivity — κ in Fourier's law of heat conduction
• Dielectric constant — κ = ε_r in capacitor formulas
• Condition number — κ(A) of a matrix
• Compressibility — κ = −(1/V)(∂V/∂p) in thermodynamics
• Von Kármán constant — κ ≈ 0.41 in turbulent boundary layers
• Curvature parameter — κ ∈ {−1, 0, +1} in cosmology
• Kappa light chain — κ in immunology; antibodies carry either κ or λ light chains

Curvature in Geometry

Curvature measures how quickly a curve turns. For a curve parameterized by arc length s with unit tangent vector T:

κ = |dT/ds|

A straight line has κ = 0, and a circle of radius R has constant curvature κ = 1/R, so the radius of curvature of any curve at a point is R = 1/κ. For the graph of a function y = f(x):

κ = |f″(x)| / (1 + f′(x)²)^(3/2)

Surfaces have two principal curvatures, κ₁ and κ₂, at each point. Their product is the Gaussian curvature K = κ₁κ₂, which Gauss proved depends only on distances measured within the surface, and their average is the mean curvature H = (κ₁ + κ₂)/2, which is zero for soap films.

Cohen's Kappa in Statistics

Cohen's kappa measures how well two raters agree when classifying items into categories, correcting for the agreement that would occur by chance alone:

κ = (p_o − p_e) / (1 − p_e)

where p_o is the observed proportion of agreement and p_e is the proportion expected by chance, computed from each rater's marginal frequencies. κ = 1 means perfect agreement, κ = 0 means agreement no better than chance, and negative values mean worse than chance. Jacob Cohen introduced the statistic in 1960. A widely quoted interpretation scale by Landis and Koch reads:

Variants include weighted kappa for ordinal categories, where a disagreement of one category counts less than a disagreement of three, and Fleiss' kappa for more than two raters. The statistic is standard in medical diagnosis studies, content analysis, and the evaluation of machine-learning labels against human annotators.

Thermal Conductivity and the Dielectric Constant

Fourier's law of heat conduction says that heat flux is proportional to the temperature gradient:

q = −κ ∇T

Thermal conductivity κ (many engineering texts use k) is measured in W/(m·K). Copper conducts at about 400 W/(m·K), glass at about 1, and air at 0.026, which is why insulating materials trap pockets of air.

The dielectric constant κ (also written ε_r) is the factor by which a material increases a capacitor's capacitance when it fills the gap between the plates:

C = κ ε₀ A / d

Vacuum has κ = 1, paper about 3, and water about 80. Introductory physics textbooks favor κ, while electromagnetism texts prefer ε_r; the two are the same number.

Condition Number in Numerical Analysis

The condition number κ(A) of a matrix tells you how much a small error in the input of a linear system Ax = b can be amplified in the solution:

κ(A) = ‖A‖ · ‖A⁻¹‖

For the 2-norm this equals the ratio of the largest to the smallest singular value. A matrix with κ near 1 is well-conditioned; a matrix with κ of 10¹⁰ loses about ten digits of accuracy, and computations in double precision become unreliable. Preconditioning and regularization are ways of lowering κ.

Kappa in Different Fields

Mathematics

• Curvature κ of curves and principal curvatures κ₁, κ₂ of surfaces
• Condition number κ(A) of a matrix or problem
• Cardinal numbers, where κ and λ are the traditional names for infinite cardinals in set theory

Physics

• Thermal conductivity κ in Fourier's law
• Dielectric constant κ of a capacitor's insulating material
• Curvature parameter κ in the Friedmann equations, which decides whether the universe is closed (+1), flat (0), or open (−1)
• Extinction coefficient κ in optics, which sets how quickly light is absorbed in a material

Engineering

• Von Kármán constant κ ≈ 0.41 in the logarithmic law of the wall for turbulent flow near a surface
• Curvature in road design and in beam bending, where the bending moment is M = EIκ
• Thermal conductivity in heat-exchanger and insulation design

Statistics and Biology

• Cohen's kappa, weighted kappa, and Fleiss' kappa for inter-rater agreement
• Kappa (κ) and lambda (λ) light chains of antibodies; the κ/λ ratio is a clinical test for blood disorders
• Kappa-opioid receptor, one of the three classical opioid receptors

How to Type Kappa

Windows: Kappa has no classic Alt code, so Alt + 954 works only in Microsoft Word and other apps that accept Unicode Alt codes. In Word, the most reliable method is to type 03BA and press Alt + X. In other programs, open Character Map, search for "kappa," and copy the character, or use the emoji panel (Win + .) and its Symbols tab.

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

iPhone and iPad: Go to Settings > General > Keyboard > Keyboards > Add New Keyboard and choose Greek. Switch keyboards with the globe key; κ sits on the K 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 "kappa."

HTML: Write κ for κ, Κ for Κ, and ϰ (or ϰ) for ϰ.

LaTeX: In math mode, \kappa gives κ. The cursive \varkappa requires the amssymb package. There is no \Kappa; for an upright capital use \mathrm{K}. The following document compiles as written:

1\documentclass{article}
2\usepackage{amsmath,amssymb}
3\begin{document}
4Curvature is $\kappa$, the cursive form is $\varkappa$, and capital kappa is $\mathrm{K}$.
5\[
6  \kappa = \frac{|f''(x)|}{\left(1 + f'(x)^2\right)^{3/2}},
7  \qquad
8  \kappa = \frac{p_o - p_e}{1 - p_e},
9  \qquad
10  \mathbf{q} = -\kappa \nabla T,
11  \qquad
12  \kappa(A) = \|A\| \, \|A^{-1}\|
13\]
14\end{document}
Copy

For kappa in running text, the textgreek package provides \textkappa, which gives an upright κ that matches the surrounding font.

Related Greek Letters

Curvature κ is the reciprocal of the radius of curvature ρ; the dielectric constant κ is another name for ε_r; and κ keeps company with λ both as antibody light chains and as infinite cardinals.

• Lambda symbol (Λ λ) — the other antibody light chain and the partner cardinal
• Epsilon symbol (Ε ε) — permittivity ε, the other name for the dielectric constant
• Rho symbol (Ρ ρ) — the radius of curvature ρ = 1/κ
• Sigma symbol (Σ σ) — the singular values whose ratio is the condition number

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

• Kappa 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
• Iota symbol (ι)
• Lambda symbol (λ)

FAQ

What does the kappa symbol mean?

In mathematics κ is usually curvature or the condition number of a matrix; in statistics it is Cohen's kappa, a measure of agreement between raters; in physics it is thermal conductivity or the dielectric constant. Outside science, "Kappa" is a Twitch emote that signals sarcasm.

What is a good Cohen's kappa value?

By the Landis and Koch scale, κ above 0.60 is "substantial" agreement and above 0.80 is "almost perfect." Values below 0.40 suggest the raters or the category definitions need work. The threshold that counts as acceptable depends on the field; medical coding studies often require κ ≥ 0.70.

What is the difference between κ and ϰ?

Both are lowercase kappa. The cursive ϰ (LaTeX \varkappa from amssymb, Unicode U+03F0) is a typographic variant that some authors use to avoid confusion with a nearby κ or with the Latin k. They have no different meaning by themselves.

Is Κ the same as the letter K?

No. Uppercase kappa Κ (U+039A) and the Latin K (U+004B) look identical in nearly every font but are different characters. Software treats them as unrelated, which matters for search, spell-checking, and code, so use the Latin K unless you specifically need the Greek letter.

What is the kappa Greek letter?

Kappa is the 10th letter of the Greek alphabet, written Κ (uppercase) and κ (lowercase), with a cursive variant ϰ. It sounds like the English "k", is pronounced "KAP-uh", and stood for 20 in the Greek numeral system.

How do you copy and paste the kappa symbol?

Copy κ, Κ or ϰ from the box at the top of this page. In HTML write κ, Κ or ϰ; in Word type 03BA and press Alt + X; on a Mac switch to the Greek input source and press K.

How do you write kappa in LaTeX?

In math mode use $\kappa$ for κ and $\varkappa$ (with \usepackage{amssymb}) for ϰ. Uppercase kappa is identical to the Latin K, so LaTeX has no \Kappa; write $\mathrm{K}$. In running text, \textkappa from the textgreek package gives an upright κ.

Writing with Greek Symbols

Papers that mix curvatures, agreement statistics, and conductivities need kappa typeset cleanly and consistently. In LaTeX that means \kappa in math mode rather than a pasted Unicode character, which keeps spacing and italics correct throughout the document.

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