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an SI unit of plane angle; the angle between two radii of a circle that cut off on the circumference an arc equal in length to the radius. 1 radian is equivalent to 57.296 degrees and π/2 radians equals a right angle
Collins Discovery Encyclopedia, 1st edition © HarperCollins Publishers 2005

(ray -dee-ăn) A unit of angle used in plane geometry that is the angle subtended at the center of a circle by an arc of the circle equal in length to the circle's radius. Thus 2π radians equals 360°, 1 radian equals 57°.296, and 1° equals 0.017 radians. Solid angles, used in spherical geometry, are measured in steradians.
Collins Dictionary of Astronomy © Market House Books Ltd, 2006
The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.

the central angle subtended in a circle by an arc whose length is equal to the circle’s radius. A radian is approximately equal to 57°17′44.8″. The radian is used as the unit of measurement of angles in what is known as circular, or radian, measure. If the circular measure of an angle is a radians, the angle contains (180a/π)°. Conversely, if the degree measure of an angle is , the circular measure is πn/180 radians. For example, angles of 30°, 45°, 60°, 90°, and 180° correspond to angles of π/6, π/4, π/3, π/2, and π radians, respectively.