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Degree—Wolfram Documentation

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Degree

gives the number of radians in one degree. It has a numerical value of .

Details
Details and Options Details and Options
Background & Context
Examples  
Basic Examples  
Scope  
Applications  
Properties & Relations  
Possible Issues  
Neat Examples  
See Also
Tech Notes
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Degree

Degree

gives the number of radians in one degree. It has a numerical value of .

Details

Background & Context

  • Degree is the symbol representing the number of radians in one angular degree (1°), which can also be input into the Wolfram Language as ∖[Degree]. Degree has exact value and numerical value . The use of Degree is especially common in calculations involving plane geometry and trigonometry. While most angle-related functions in the Wolfram Language take radian measures as their arguments and return radian measures as results, the symbol Degree can be used as a multiplier when entering values in degree measures (e.g. Cos[30 Degree]).
  • When Degree is used as a symbol, it is propagated as an exact quantity that can be expressed in terms of Pi using FunctionExpand. While many expressions involving Degree (e.g. Cos[30 Degree]) are automatically expanded in terms of simpler functions, expansion and simplification of more complicated expressions involving Degree (e.g. Cos[12 Degree]) may require use of functions such as FunctionExpand and FullSimplify.
  • As a result of its close relationship with Pi, Degree is known to be both irrational and transcendental, meaning it can be expressed neither as a ratio of integers nor as the root of any integer polynomial. While (like Pi) it is not known if Degree is normal (meaning the digits in its base- expansion are equally distributed) to any base, its known digits are very uniformly distributed.
  • Degree can be evaluated to arbitrary numerical precision using N. In fact, calculating the first million decimal digits of Degree takes only a fraction of a second on a modern desktop computer due to the rapid convergence of the Chudnovsky formula for Pi. RealDigits can be used to return a list of digits of Degree and ContinuedFraction to obtain terms of its continued fraction expansion.
  • Note that the Wolfram Language unit framework represents the angular degree as Quantity[1,"AngularDegrees"] for the purposes of unit encoding and conversion. Some care is therefore needed in distinguishing the mathematical constant Degree (which relates degrees to radians) and the unit of angular measure itself. This slightly unfortunate dual nature of degrees has its roots in the treatment of angular measure by the SI system of units, which treats radians as a dimensionless measure whose explicit specification is optional.

Examples

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Basic Examples  (3)

Use in arguments of trigonometric functions:

Wolfram Language code: Cos[30 Degree]

Evaluate to any precision:

Wolfram Language code: N[Degree, 50]

Degree can be entered as deg:

Wolfram Language code: °

Scope  (1)

Do an exact numerical computation:

Wolfram Language code: Round[Pi / Degree ^ 2]

Applications  (1)

Plot a triangle with angles specified in degrees:

Wolfram Language code: With[{α = 30 °, β = 75°}, Graphics[Polygon[{{0, 0}, {1, 0}, {Cos[α] Csc[α + β] Sin[β], Csc[α + β] Sin[α] Sin[β]}}]]]

Properties & Relations  (4)

Evaluate symbolic relations involving Degree:

Wolfram Language code: Pi / 4 < 60Degree < Pi

Use FullSimplify to simplify expressions containing Degree:

Wolfram Language code: FullSimplify[Pi / Degree]
Wolfram Language code: FullSimplify[Degree - 4ArcTan[1 / 5] + ArcTan[1 / 239]]

Degree is not an algebraic number:

Wolfram Language code: Element[Degree, Algebraics]

Radians may be converted to degrees using UnitConvert and Quantity as well as Degree:

Wolfram Language code: UnitConvert[Quantity[180, "AngularDegrees"], "Radians"]
Wolfram Language code: 180 Degree//FullSimplify

Possible Issues  (1)

Degree is typically not generated as output:

Wolfram Language code: ArcSin[1] / 90

FullSimplify does not transform expressions to Degree:

Wolfram Language code: FullSimplify[%]

Neat Examples  (1)

Form degree graphics of various degrees of complexity:

Wolfram Language code: Nest[Binomial[#, # ^ #]&, Degree, 3]//TraditionalForm
Wolfram Language code: N[%]
Wolfram Language code: Nest[Subscript[#, #] ^ #&, Degree, 6]//TraditionalForm
Wolfram Language code: Table[Nest[# ^ (#) ^ #&, Degree, RandomInteger[{1, 3}]], {4}, {4}]//TraditionalForm
Wolfram Language code: Det[%]
Wolfram Language code: N[%]
Wolfram Research (1988), Degree, Wolfram Language function, https://reference.wolfram.com/language/ref/Degree.html (updated 2002).

Text

Wolfram Research (1988), Degree, Wolfram Language function, https://reference.wolfram.com/language/ref/Degree.html (updated 2002).

CMS

Wolfram Language. 1988. "Degree." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2002. https://reference.wolfram.com/language/ref/Degree.html.

APA

Wolfram Language. (1988). Degree. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Degree.html

BibTeX

@misc{reference.wolfram_2026_degree, author="Wolfram Research", title="{Degree}", year="2002", howpublished="\url{https://reference.wolfram.com/language/ref/Degree.html}", note=[Accessed: 14-November-2026]}

BibLaTeX

@online{reference.wolfram_2026_degree, organization={Wolfram Research}, title={Degree}, year={2002}, url={https://reference.wolfram.com/language/ref/Degree.html}, note=[Accessed: 14-November-2026]}