A degree is the name of several units of measurement, and also an academic qualification.
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A degree of arc, usually symbolized by the symbol °, is a measurement of plane angles, or of a location along a great circle of a sphere (such as the Earth or the celestial sphere); it represents going one-360th the way around the circumference of a circle or sphere. Where degree measures are used, you'll also find minutes (one-60th of a degree, or 1/21600 of the way around, symbolized by a single quotation mark) and seconds (one-60ths of a minute, or 1/1296000 of the way around , symbolized by a double quotation mark). Thus 40°12'13" is 40 degrees, 12 minutes, 13 seconds, or about 11/100 of the way around.
In more mathematical context, angles are usually measured in radians instead of degrees. One degree equals π/180 radians.
There was also an attempt to introduce a decimal degree, see gon.
SI unit of temperature, the kelvin, uses the symbol K and is not proceeded by "degrees" or the degree sign (°).
Mathematics and Physics
A "degree of freedom" is a concept in mathematics, physics and engineering. See the article "degrees of freedom" for the use of this concept.
The degree of a term of a polynomial is the exponent on the variable in that term; the degree of an entire polynomial is the degree of the term of highest degree. For example, the term of highest degree in 2x3 + 4x2 + x + 7 is 2x3; this term, and therfore the entire polynomial, are said to have degree 3. For polynomials in two or more variables, the degree of a term is the sum of the exponents of the variables in the term, and the degree of a polynomial is again the degree of the term of highest degree. For example, the degree of the polynomial x2y2 + 3x3 + 4y is 4, the degree of the x2y2 term.
A degree is an award conferred on a person by an institution of higher education, generally in recognition of academic achievement, but in some cases merely as a honor for non-academic work (the later are called honorary degrees, or degrees honoris causa). A person issued a degree is permitted to place initials after their name indicating the award, and in some cases (Doctorates and Higher Doctorates) assume the title of Doctor (by custom people with doctorates honoris causa do not assume this title).
In music or music theory a scale degree is an individual note of a scale. First, second, third, fourth, fifth, sixth, and seventh may be used to indicate degree as well as tonic, supertonic, mediant, subdominant, dominant, submediant, subtonic, and leading tone. The degree sign "°" is used to indicate that a chord is diminished. The chord built upon a the fifth scale degree, or dominant, with also be called the fifth or dominant chord.