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.
