An angle (from the Lat. angulus, a corner, a diminutive, of which the
primitive form, angus) is the figure formed by two rays sharing a common
endpoint, called the vertex of the angle. Angles provide a means of expressing
the difference in slope between two rays meeting at a vertex without the need to
explicitly define the slopes of the two rays. Angles are studied in geometry and
Euclid defines a plane angle as the inclination to each other, in a plane, of
two lines which meet each other, and do not lie straight with respect to each
other. According to Proclus an angle must be either a quality or a quantity, or
a relationship. The first concept was used by Eudemus, who regarded an angle as
a deviation from a straight line; the second by Carpus of Antioch, who regarded
it as the interval or space between the intersecting lines;
Euclid adopted the
third concept, although his definitions of right, acute, and obtuse angles are
Units of measure for angles
In order to measure an angle, a circle centered at the vertex is drawn. Since
the circumference of a circle is always directly proportional to the length of
its radius, the measure of the angle is independent of the size of the circle.
Note that angles are dimensionless, since they are defined as the ratio of
- The degree measure of the angle is the length of the arc, divided
by the circumference of the circle, and multiplied by 360. The symbol for
degrees is a small superscript circle, as in 360ï¿½. 2π radians is equal to
360ï¿½ (a full circle), so one radian is about 57ï¿½ and one degree is π/180
- The radian measure of the angle is the length of the arc cut out
by the angle, divided by the circle's radius. The SI system of units uses
radians as the (derived) unit for angles.
- The grad, also called grade or gon, is an angular
measure where the arc is divided by the circumference, and multiplied by
400. It is used mostly in triangulation.
- The point is used in navigation, and is defined as 1/32 of a
circle, or exactly 11.25ï¿½.
- The full circle or full turns represents the number or
fraction of complete full turns. For example, π/2 radians = 90ï¿½ = 1/4 full
Conventions on measurement
A convention universally adopted in mathematical writing is that angles given
a sign are positive angles if measured counterclockwise, and negative angles if
measured clockwise, from a given line. If no line is specified, it can be
assumed to be the x-axis in the Cartesian plane. In navigation and other areas
this convention may not be followed.
In mathematics radians are assumed unless specified otherwise because this
removes the arbitrariness of the number 360 in the degree system and because the
trigonometric functions can be developed into particularly simple Taylor series
if their arguments are specified in radians.