Angle measurement encompasses both direct physical measurement using a measuring instrument such as a
protractor, as well as the theoretical calculation of angle size from other known quantities. While the measurement of angles is intrinsically linked with rotation and circles, there are various perspectives as to exactly
what is being measured, including amongst others: the amount of rotation about the vertex of one ray to the other; the amount of opening between the rays; or the length of the arc that subtends the angle at the centre of a unit circle. Angles of special significance (such as the right angle) inform the systems and units of angular measurement, which is not the case for length where the units of measurement (metres, feet) are arbitrary. Broadly there are two approaches to measuring angles: relative to a reference angle (such as a right angle); and circular measurement.
Reference angle A chosen reference angle (right angle, straight angle or full angle) can be divided into equal parts, and the size of one part used as a unit for measurement of other angles. In the most common method of practical angular measurement a right angle is divided into 90 equal parts called
degrees, while in the rarely used
centesimal system, a right angle is divided into 100 equal parts called
gradians. while the ratio of length s by the circumference C is the number of
turns: \theta = \frac{s}{r} \, \mathrm{rad} = \frac{s}{C} \, \mathrm{turn} = \frac{s}{2\pi r} \, \mathrm{turn} The value of thus defined is independent of the size of the circle: if the length of the radius is changed, then both the circumference and the arc length change in the same proportion, so the ratios and are unaltered. The ratio is called the "radian measure" of an angle, but is also used to define a
unit of measurement called a radian, which is defined as an angle for which the ratio = 1. and in particular, the
radian is defined as dimensionless in the
International System of Units. This convention prevents angles providing information for
dimensional analysis. For example, when one measures an angle in radians by dividing the arc length by the radius, one is essentially dividing a length by another length, and the units of length cancel each other out. Therefore the result—the angle—doesn't have a physical "dimension" like meters or seconds. This holds true with all angle units, such as radians, degrees, or turns—they all represent a pure number quantifying how much something has turned. This is why, in many equations, angle units seem to "disappear" during calculations, which feels inconsistent and can lead to mixing up angle units. This has led to significant discussion among scientists and educators. Some scientists have suggested treating the angle as having its own fundamental dimension, similar to length or time. This would mean that angle units like radians would always be explicitly present in calculations, facilitating dimensional analysis. However, this approach would also require changing many well-known mathematical and physics formulas, making them longer and perhaps a bit less familiar. For now, the established practice is to write angle units where appropriate but consider them dimensionless, understanding that these units are important but behave differently from meters or kilograms.
Signed angles , angles on the
unit circle count as positive in the
counterclockwise direction, and negative in the
clockwise direction. An angle denoted as might refer to any of four angles: the clockwise angle from B to C about A, the anticlockwise angle from B to C about A, the clockwise angle from C to B about A, or the anticlockwise angle from C to B about A, It is therefore frequently helpful to impose a convention that allows positive and negative angular values to represent
orientations and/or
rotations in opposite directions or "sense" relative to some reference. In a two-dimensional
Cartesian coordinate system, an angle is typically defined by its two sides, with its vertex at the origin. The
initial side is on the positive
x-axis, while the other side or
terminal side is defined by the measure from the initial side in radians, degrees, or turns, with
positive angles representing rotations toward the positive
y-axis and
negative angles representing rotations toward the negative
y-axis. When Cartesian coordinates are represented by
standard position, defined by the
x-axis rightward and the
y-axis upward, positive rotations are
anticlockwise, and negative cycles are
clockwise. In many contexts, an angle of −
θ is effectively equivalent to an angle of "one full turn minus
θ". For example, an orientation represented as −45° is effectively equal to an orientation defined as or 315°. Although the final position is the same, a physical rotation (movement) of −45° is not the same as a rotation of 315° (for example, the rotation of a person holding a broom resting on a dusty floor would leave visually different traces of swept regions on the floor). In three-dimensional geometry, "clockwise" and "anticlockwise" have no absolute meaning, so the direction of positive and negative angles must be defined in terms of an
orientation, which is typically determined by a
normal vector passing through the angle's vertex and perpendicular to the plane in which the rays of the angle lie. In
navigation,
bearings or
azimuth are measured relative to north. By convention, viewed from above, bearing angles are positive clockwise, so a bearing of 45° corresponds to a north-east orientation. Negative bearings are not used in navigation, so a north-west orientation corresponds to a bearing of 315°.
Equivalent angles • Angles that have the same measure (i.e., the same magnitude) are said to be
equal or
congruent. An angle is defined by its measure and is not dependent upon the lengths of the sides of the angle (e.g., all
right angles are equal in measure). • Two angles that share terminal sides, but differ in size by an integer multiple of a turn, are called
coterminal angles. • The
reference angle (sometimes called
related angle) for any angle
θ in standard position is the positive acute angle between the terminal side of
θ and the x-axis (positive or negative). Procedurally, the magnitude of the reference angle for a given angle may determined by taking the angle's magnitude
modulo turn, 180°, or radians, then stopping if the angle is acute, otherwise taking the supplementary angle, 180° minus the reduced magnitude. For example, an angle of 30 degrees is already a reference angle, and an angle of 150 degrees also has a reference angle of 30 degrees (). Angles of 210° and 510° correspond to a reference angle of 30 degrees as well (, whose supplementary angle is 30°).
Related quantities For an angular unit, it is definitional that the
angle addition postulate holds, however some measurements or quantities related to angles are in use that do not satisfy this postulate: • The
slope or
gradient is equal to the
tangent of the angle and is often expressed as a percentage ("rise" over "run"). For very small values (less than 5%), the slope of a line is approximately the measure in radians of its angle with the horizontal direction. An
elevation grade is a slope used to indicate the steepness of roads, paths and railway lines. • The
spread between two lines is defined in
rational geometry as the square of the sine of the angle between the lines. As the sine of an angle and the sine of its supplementary angle are the same, any angle of rotation that maps one of the lines into the other leads to the same value for the spread between the lines. • Although done rarely, one can report the direct results of
trigonometric functions, such as the
sine of the angle. == Angles between curves ==