Angle

Given two intersecting lines or line segments, the amount of rotation about the point of intersection (the vertex) required to bring one into correspondence with the other is called the angle between them. The term "plane angle" is sometimes used to distinguish angles in a plane from solid angles measured in space (International Standards Organization 1982, p. 5).

The term "angle" can also be applied to the rotational offset between intersecting planes about their common line of intersection, in which case the angle is called the dihedral angle of the planes.

Angles are usually measured in degrees (denoted ), radians (denoted rad, or without a unit), or sometimes gradians (denoted grad).

The concept of an angle can be generalized from the circle to the sphere, in which case it is known as solid angle. The fraction of a sphere subtended by an object (its solid angle) is measured in steradians, with the entire sphere corresponding to steradians.

One full rotation in these three measures corresponds to , rad, or 400 grad. Half a full rotation is called a straight angle, and a quarter of a full rotation is called a right angle. An angle less than a right angle is called an acute angle, an angle greater than a right angle (but less than a straight angle) is called an obtuse angle, and an angle greater than a straight angle (but less than a full angle) is called a reflex angle.

The use of degrees to measure angles harks back to the Babylonians, whose sexagesimal number system was based on the number 60. likely arises from the Babylonian year, which was composed of 360 days (12 months of 30 days each). The degree is further divided into 60 arc minutes, and an arc minute into 60 arc seconds. A more natural measure of an angle is the radian. It has the property that the arc length around a circle is simply given by the radian angle measure times the circle radius. The radian is also the most useful angle measure in calculus because the derivative of trigonometric functions such as

does not require the insertion of multiplicative constants like . Gradians are sometimes used in surveying (they have the nice property that a right angle is exactly 100 gradians), but are encountered infrequently, if at all, in mathematics.

A ruled semicircle used for measuring and drawing angles is called a protractor. A compass can also be used to draw circular arcs of some angular extent.

Wolfram Web Resources