# Angles

âˆ , the angle symbol

In geometry, an angle is the figure formed by two rays sharing a common endpoint, called the vertex of the angle.[1] The magnitude of the angle is the “amount of rotation” that separates the two rays, and can be measured by considering the length of circular arc swept out when one ray is rotated about the vertex to coincide with the other (see “Measuring angles”, below). Where there is no possibility of confusion, the term “angle” is used interchangeably for both the geometric configuration itself and for its angular magnitude (which is simply a numerical quantity).

The word angle comes from the Latin word angulus, meaning “a corner”. The word angulus is a diminutive, of which the primitive form, angus, does not occur in Latin. Cognate words are the Greek á¼€Î³ÎºÏÎ»Î¿Ï‚ (ankylÎ¿s), meaning “crooked, curved,” and the English word “ankle“. Both are connected with the Proto-Indo-European root *ank-, meaning “to bend” or “bow”.[2]

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 certainly quantitative.[3]

## Measuring angles

Two angles are sometimes called congruent if there exists an isometry that transforms one of the angles into the other angle. The size of an angle is normally characterized by the smallest positive rotation that maps one of the rays into the other. Two angles are congruent if and only if they correspond to the same (smallest positive) rotation. Thus an angle as two rays is characterized by an angle of rotation. To avoid confusion when no isometry exists between particular representations of angles, angles that Euclid called “equal” are described as “equal in measure”.

In many geometrical situations, angles that differ by an exact multiple of a full circle are effectively equivalent (it makes no difference how many times a line is rotated through a full circle because it always ends up in the same place). However, this is not always the case. For example, when tracing a curve such as a spiral using polar coordinates, an extra full turn gives rise to a quite different point on the curve.

The measure of angle Î¸ is the quotient of s and r.

In order to measure an angle Î¸, a circular arc centered at the vertex of the angle is drawn, e.g. with a pair of compasses. The length of the arc s is then divided by the radius of the circle r, and possibly multiplied by a scaling constant k (which depends on the units of measurement that are chosen):

$\theta = k \frac{s}{r}.$

The value of Î¸ thus defined is independent of the size of the circle: if the length of the radius is changed then the arc length changes in the same proportion, so the ratio s/r is unaltered.

### Units

In dimensional analysis, angles are considered to be dimensionless. There are several units used to measure angles. Of these units, listed below according to magnitude, the degree and the radian are by far the most common.

With the notable exception of the radian, most units of angular measurement are defined such that one full circle (i.e. one turn) is equal to n units, for some whole number n. For example, in the case of degrees, n = 360. A full circle of n units is obtained by setting k = n/(2Ï€) in the formula above. (Proof. The formula above can be rewritten as k = Î¸r/s. One turn, for which Î¸ = n units, corresponds to an arc equal in length to the circle’s circumference, which is 2Ï€r, so s = 2Ï€r. Substituting n for Î¸ and 2Ï€r for s in the formula, results in k = nr/(2Ï€r) = n/(2Ï€).)

• The turn (or full circle, revolution, rotation, or cycle) is one full circle. A turn can be subdivided in centiturns and milliturns. A turn is abbreviated Ï„ or rev or rot depending on the application, but just r in rpm (revolutions per minute). 1 turn = 360Â° = 2Ï€ rad = 400 gon = 4 right angles.
• The quadrant is 1/4 of a turn, i.e. a right angle. It is the unit used in Euclid’s Elements. 1 quad. = 90Â° = Ï€/2 rad = 1/4 turn = 100 gon. In German the symbol âˆŸ has been used to denote a quadrant.
• The angle of the equilateral triangle is 1/6 of a turn. It was the unit used by the Babylonians[citation needed], and is especially easy to construct with ruler and compasses. The degree, minute of arc and second of arc are sexagesimal subunits of the Babylonian unit. 1 Babylonian unit = 60Â° = Ï€/3 rad â‰ˆ 1.047197551 rad.

• The astronomical hour angle is 1/24 of a turn. Since this system is amenable to measuring objects that cycle once per day (such as the relative position of stars), the sexagesimal subunits are called minute of time and second of time. Note that these are distinct from, and 15 times larger than, minutes and seconds of arc. 1 hour = 15Â° = Ï€/12 rad = 1/6 quad. = 1/24 turn â‰ˆ 16.667 gon.
• The point, used in navigation, is 1/32 of a turn. 1 point = 1/8 of a right angle = 11.25Â° = 12.5 gon. Each point is subdivided in four quarter-points so that 1 turn equals 128 quarter-points.
• Eratosthenes used a unit of 6Â° so that a whole turn was divided in 60 units.
• The Babylonians sometimes used the unit pechus of about 2Â° or 2Â½Â°.
• The binary degree, also known as the binary radian (or brad), is 1/256 of a turn.[4] The binary degree is used in computing so that an angle can be efficiently represented in a single byte (albeit to limited precision). Other measures of angle used in computing may be based on dividing one whole turn into 2n equal parts for other values of n.[5]
• The degree, denoted by a small superscript circle (Â°), is 1/360 of a turn, so one full circle is 360Â°. One advantage of this old sexagesimal subunit is that many angles common in simple geometry are measured as a whole number of degrees. Fractions of a degree may be written in normal decimal notation (e.g. 3.5Â° for three and a half degrees), but the following sexagesimal subunits of the “degree-minute-second” system are also in use, especially for geographical coordinates and in astronomy and ballistics:
• The grad, also called grade, gradian, or gon, is 1/400 of a turn, so a right angle is 100 grads. It is a decimal subunit of the quadrant. A kilometre was historically defined as a centi-gon of arc along a great circle of the Earth, so the kilometre is the decimal analog to the sexagesimal nautical mile. The gon is used mostly in triangulation.
• The minute of arc (or MOA, arcminute, or just minute) is 1/60 of a degree = 1/3600 turn. It is denoted by a single prime ( â€² ). For example, 3Â° 30â€² is equal to 3 + 30/60 degrees, or 3.5 degrees. A mixed format with decimal fractions is also sometimes used, e.g. 3Â° 5.72â€² = 3 + 5.72/60 degrees. A nautical mile was historically defined as a minute of arc along a great circle of the Earth.
• The mil is approximately equal to a milliradian. There are several definitions.
• The second of arc (or arcsecond, or just second) is 1/60 of a minute of arc and 1/3600 of a degree. It is denoted by a double prime ( â€³ ). For example, 3Â° 7â€² 30â€³ is equal to 3 + 7/60 + 30/3600 degrees, or 3.125 degrees.

### Positive and negative angles

In mathematics, the angle from the first to the second coordinate axis of a coordinate system is considered as positive. Therefore angles given a sign are positive angles if measured anticlockwise, and negative angles if measured clockwise, from a given line. If no line is specified, it can be assumed to be the first coordinate axis (x-axis) in the Cartesian plane. In many geometrical situations a negative angle of âˆ’Î¸ is effectively equivalent to a positive angle of “one full turn less Î¸“. For example, a clockwise rotation of 45Â° (that is, an angle of âˆ’45Â°) is often effectively equivalent to an anticlockwise rotation of 360Â° âˆ’ 45Â° (that is, an angle of 315Â°).

In three dimensional geometry, “clockwise” and “anticlockwise” have no absolute meaning, so the direction of positive and negative angles must be defined relative to some reference, which is typically a vector passing through the angle’s vertex and perpendicular to the plane in which the rays of the angle lie.

In navigation, bearings are measured from north, increasing clockwise, so a bearing of 45 degrees is north-east. Negative bearings are not used in navigation, so north-west is 315 degrees.

### Alternative ways of measuring the size of an angle

There are several alternatives to measuring the size of an angle by the corresponding angle of rotation. The grade of a slope, or gradient is equal to the tangent of the angle, or sometimes the sine. Gradients are often expressed as a percentage. For very small values (less than 5%), the grade of a slope is approximately the measure of an angle in radians.

In rational geometry the spread between two lines is defined at the square of sine of the angle between the lines. Since 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 of the spread between the lines.

### Astronomical approximations

Astronomers measure angular separation of objects in degrees from their point of observation.

• 1Â° is approximately the width of a little finger at arm’s length.
• 10Â° is approximately the width of a closed fist at arm’s length.
• 20Â° is approximately the width of a handspan at arm’s length.

These measurements clearly depend on the individual subject, and the above should be treated as rough approximations only.

## Identifying angles

In mathematical expressions, it is common to use Greek letters (Î±, Î², Î³, Î¸, Ï†, …) to serve as variables standing for the size of some angle. (To avoid confusion with its other meaning, the symbol Ï€ is typically not used for this purpose.) Lower case roman letters (a, b, c, …) are also used. See the figures in this article for examples.

In geometric figures, angles may also be identified by the labels attached to the three points that define them. For example, the angle at vertex A enclosed by the rays AB and AC (i.e. the lines from point A to point B and point A to point C) is denoted âˆ BAC or BÃ‚C. Sometimes, where there is no risk of confusion, the angle may be referred to simply by its vertex (“angle A”).

Potentially, an angle denoted, say, âˆ BAC might refer to any of four angles: the clockwise angle from B to C, the anticlockwise angle from B to C, the clockwise angle from C to B, or the anticlockwise angle from C to B, where the direction in which the angle is measured determines its sign (see Positive and negative angles). However, in many geometrical situations it is obvious from context that the positive angle less than or equal to 180 degrees is meant, and no ambiguity arises. Otherwise, a convention may be adopted so that âˆ BAC always refers to the anticlockwise (positive) angle from B to C, and âˆ CAB to the anticlockwise (positive) angle from C to B.

## Types of angles

 The complementary angles a and b (b is the complement of a, and a is the complement of b). Acute (a), obtuse (b), and straight (c) angles. Here, a and b are supplementary angles.
• An angle equal to 1/4 turn (90Â° or Ï€/2 radians) is called a right angle.
Two lines that form a right angle are said to be perpendicular or orthogonal.
• Angles equal to 1/2 turn (180Â° or two right angles) are called straight angles.
• Angles that are not right angles or a multiple of a right angle are called oblique angles.
• Angles smaller than a right angle (less than 90Â°) are called acute angles (“acute” meaning “sharp”).
• Angles larger than a right angle and smaller than a straight angle (between 90Â° and 180Â°) are called obtuse angles (“obtuse” meaning “blunt”).
• Angles larger than a straight angle but less than 1 turn (between 180Â° and 360Â°) are called reflex angles.
• Angles that have the same measure (i.e. the same magnitude) are said to be congruent. Following this definition for congruent angles, 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 congruent).
• Two angles opposite each other, formed by two intersecting straight lines that form an “X”-like shape, are called vertical angles or opposite angles or vertically opposite angles. These angles are equal in measure.
• Angles that share a common vertex and edge but do not share any interior points are called adjacent angles.
• Two angles that sum to one right angle (90Â°) are called complementary angles.
The difference between an angle and a right angle is termed the complement of the angle.
• Two angles that sum to a straight angle (180Â°) are called supplementary angles.
The difference between an angle and a straight angle (180Â°) is termed the supplement of the angle.
• Two angles that sum to one turn (360Â°) are called explementary angles or conjugate angles.
• An angle that is part of a simple polygon is called an interior angle if it lies on the inside of that simple polygon. A concavesimple polygon has at least one interior angle that exceeds 180Â°.
In Euclidean geometry, the measures of the interior angles of a triangle add up to Ï€ radians, or 180Â°, or 1/2 turn; the measures of the interior angles of a simple quadrilateral add up to 2Ï€ radians, or 360Â°, or 1 turn. In general, the measures of the interior angles of a simple polygon with n sides add up to [(n âˆ’ 2) Ã— Ï€] radians, or [(n âˆ’ 2) Ã— 180]Â°, or (2n âˆ’ 4) right angles, or (n/2 âˆ’ 1) turn.
• The angle supplementary to the interior angle is called the exterior angle. It measures the amount of rotation one has to make at this vertex to trace out the polygon. If the corresponding interior angle is a reflex angle, the exterior angle should be considered negative. Even in a non-simple polygon it may be possible to define the exterior angle, but one will have to pick an orientation of the plane (or surface) to decide the sign of the exterior angle measure.
In Euclidean geometry, the sum of the exterior angles of a simple polygon will be one full turn (360Â°).
• Some authors use the name exterior angle of a simple polygon to simply mean the explementary (not supplementary!) of the interior angle.[6] This conflicts with the above usage.
• The angle between two planes (such as two adjacent faces of a polyhedron) is called a dihedral angle. It may be defined as the acute angle between two lines normal to the planes.
• The angle between a plane and an intersecting straight line is equal to ninety degrees minus the angle between the intersecting line and the line that goes through the point of intersection and is normal to the plane.
• If a straight transversal line intersects two parallel lines, corresponding (as well as alternate) angles at the two points of intersection are equal in size; adjacent angles are supplementary (that is, their measures add to Ï€ radians, or 180Â°).
• A reference angle is the acute version of any angle determined by repeatedly subtracting or adding 180 degrees, and subracting the result from 180 degrees if necessary, until a value between 0 degrees and 90 degrees is obtained. For example, an angle of 30 degrees has a reference angle of 30 degrees, and an angle of 150 degrees also has a reference angle of 30 degrees (180-150). An angle of 750 degrees has a reference angle of 30 degrees (750-720).[7]

## A formal definition

### Using trigonometric functions

A Euclidean angle is completely determined by the corresponding right triangle. In particular, if Î¸ is a Euclidean angle, it is true that

$\cos \theta = \frac{x}{\sqrt{x^2 + y^2}}$

and

$\sin \theta = \frac{y}{\sqrt{x^2 + y^2}}$

for two numbers x and y. So an angle in the Euclidean plane can be legitimately given by two numbers x and y.

To the ratio y/x there correspond two angles in the geometric range 0 < Î¸ < 2Ï€, since

$\frac{\sin \theta}{\cos \theta } = \frac{\frac{y}{\sqrt{x^2 + y^2}}}{\frac{x}{\sqrt{x^2 + y^2}}} = \frac{y}{x} = \frac{-y}{-x} = \frac{\sin (\theta + \pi)}{\cos (\theta + \pi) }.$

## Angles between curves

The angle between the two curves at P is defined as the angle between the tangents A and B at P

The angle between a line and a curve (mixed angle) or between two intersecting curves (curvilinear angle) is defined to be the angle between the tangents at the point of intersection. Various names (now rarely, if ever, used) have been given to particular cases:â€”amphicyrtic (Gr. á¼€Î¼Ï†Î¯, on both sides, ÎºÏ…ÏÏ„ÏŒÏ‚, convex) or cissoidal (Gr. ÎºÎ¹ÏƒÏƒÏŒÏ‚, ivy), biconvex; xystroidal or sistroidal (Gr. Î¾Ï…ÏƒÏ„ÏÎ¯Ï‚, a tool for scraping), concavo-convex; amphicoelic (Gr. ÎºÎ¿Î¯Î»Î·, a hollow) or angulus lunularis, biconcave.[8]

## Dot product and generalisation

In the Euclidean plane, the angle Î¸ between two vectors u and v is related to their dot product and their lengths by the formula

$\mathbf{u} \cdot \mathbf{v} = \cos(\theta)\ \|\mathbf{u}\|\ \|\mathbf{v}\|.$

This formula supplies an easy method to find the angle between two planes (or curved surfaces) from their normal vectors and between skew lines from their vector equations.

## Inner product

To define angles in an abstract real inner product space, we replace the Euclidean dot product ( Â· ) by the inner product $\langle\cdot,\cdot\rangle$, i.e.

$\langle\mathbf{u},\mathbf{v}\rangle = \cos(\theta)\ \|\mathbf{u}\|\ \|\mathbf{v}\|.$

In a complex inner product space, the expression for the cosine above may give non-real values, so it is replaced with

$Re(\langle\mathbf{u},\mathbf{v}\rangle) = \cos(\theta)\ \|\mathbf{u}\|\ \|\mathbf{v}\|.$

or, more commonly, using the absolute value, with

$|\langle\mathbf{u},\mathbf{v}\rangle| = \cos(\theta)\ \|\mathbf{u}\|\ \|\mathbf{v}\|.$

The latter definition ignores the direction of the vectors and thus describes the angle between one-dimensional subspaces $span(\mathbf{u})$ and $span(\mathbf{v})$ spanned by the vectors $\mathbf{u}$ and $\mathbf{v}$ correspondingly.

## Angles between subspaces

The definition of the angle between one-dimensional subspaces $span(\mathbf{u})$ and $span(\mathbf{v})$ given by

$|\langle\mathbf{u},\mathbf{v}\rangle| = \cos(\theta)\ \|\mathbf{u}\|\ \|\mathbf{v}\|$

in a Hilbert space can be extended to subspaces of any finite dimensions. Given two subspaces $\mathcal{U},\mathcal{W}$ with $\operatorname{dim}(\mathcal{U}):=k\leq \operatorname{dim}(\mathcal{W}):=l$, this leads to a definition of k angles called canonical or principal angles between subspaces.

## Angles in Riemannian geometry

In Riemannian geometry, the metric tensor is used to define the angle between two tangents. Where U and V are tangent vectors and gij are the components of the metric tensor G,

$\cos \theta = \frac{g_{ij}U^iV^j} {\sqrt{ \left| g_{ij}U^iU^j \right| \left| g_{ij}V^iV^j \right|}}.$

## Angles in geography and astronomy

In geography, the location of any point on the Earth can be identified using a geographic coordinate system. This system specifies the latitude and longitude of any location in terms of angles subtended at the centre of the Earth, using the equator and (usually) the Greenwich meridian as references.

In astronomy, a given point on the celestial sphere (that is, the apparent position of an astronomical object) can be identified using any of several astronomical coordinate systems, where the references vary according to the particular system. Astronomers measure the angular separation of two stars by imagining two lines through the centre of the Earth, each intersecting one of the stars. The angle between those lines can be measured, and is the angular separation between the two stars.

Astronomers also measure the apparent size of objects as an angular diameter. For example, the full moon has an angular diameter of approximately 0.5Â°, when viewed from Earth. One could say, “The Moon subtends an angle of half a degree.” The small-angle formula can be used to convert such an angular measurement into a distance/size ratio.