"∠", the angle symbol.

In geometry and trigonometry, an angle (in full, plane angle) is the figure formed by two rays sharing a common endpoint, called the vertex of the angle. 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). Also see: Angel (disambiguation) The word angle has a number of meanings. ... Image File history File links Angle_Symbol. ... Image File history File links Angle_Symbol. ... For other uses, see Geometry (disambiguation). ... Wikibooks has a book on the topic of Trigonometry The Canadarm2 robotic manipulator on the International Space Station is operated by controlling the angles of its joints. ... In Euclidean geometry, a ray, or half-line, given two distinct points A (the origin) and B on the ray, is set of points C on the line containing points A and B such that A is not strictly between C and B. O----O-----*---> A B C In geometric... An endpoint or end point is a mark of termination or completion. ... In geometry, a vertex (plural vertices) is a special kind of point, usually a corner of a polygon, polyhedron, or higher dimensional polytope. ...

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 Latin angere, meaning "to compress into a bend" or "to strangle", and the Greek ἀγκύλος (ankylοs), meaning "crooked, curved"; both are connected with the PIE root *ank-^[1], meaning "to bend" or "bow". For other uses, see Latin (disambiguation). ... Look up cognate in Wiktionary, the free dictionary. ... The Proto-Indo-European language (PIE) is the hypothetical common ancestor of the Indo-European languages, spoken by the Proto-Indo-Europeans. ...

History

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. For other uses, see Euclid (disambiguation). ... This article is about Proclus Diadochus, the Neoplatonist philosopher. ... Eudemus of Rhodes (Ευδημος) was an ancient Greek philosopher, who lived from ca. ... A line, or straight line, is, roughly speaking, an (infinitely) thin, (infinitely) long, straight geometrical object, i. ...

Measuring angles

The angle θ is the quotient of s and r.

In order to measure an angle θ, 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): Image File history File links Angle_measure. ... Image File history File links Angle_measure. ... In Euclidean geometry, an arc is a closed segment of a differentiable curve in the two-dimensional plane; for example, a circular arc is a segment of a circle. ... A beam compass and a regular compass A compass is a technical drawing instrument that can be used for inscribing circles or arcs. ...

The value of θ 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.

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. This article does not cite any references or sources. ... This article describes some of the common coordinate systems that appear in elementary mathematics. ...

Units

Angles are considered dimensionless, since they are defined as the ratio of lengths. There are, however, several units used to measure angles, depending on the choice of the constant k in the formula above. Of these units, treated in more detail below, 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 revolution) 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/(2π) in the formula above. (Proof. The formula above can be rewritten as k = θr/s. One full circle, for which θ = n units, corresponds to an arc equal in length to the circle's circumference, which is 2πr, so s = 2πr. Substituting n for θ and 2πr for s in the formula, results in k = nr/(2πr) = n/(2π).) The circumference is the distance around a closed curve. ...

• The degree, denoted by a small superscript circle (°) is 1/360 of a full circle, so one full circle is 360°. One advantage of this old sexagesimal subunit is that many angles common in simple geometry are measured as a whole number of degrees. (The problem of having all "interesting" angles measured as whole numbers is of course insolvable.) Fractions of a degree may be written in normal decimal notation (e.g. 3.5° 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 minute of arc (or MOA, arcminute, or just minute) is 1/60 of a degree. It is denoted by a single prime ( ′ ). For example, 3° 30′ is equal to 3 + 30/60 degrees, or 3.5 degrees. A mixed format with decimal fractions is also sometimes used, e.g. 3° 5.72′ = 3 + 5.72/60 degrees. A nautical mile was historically defined as a minute of arc along a great circle of the Earth.
  • 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 ( ″ ). For example, 3° 7′ 30″ is equal to 3 + 7/60 + 30/3600 degrees, or 3.125 degrees.

θ = s/r rad = 1 rad.

• The radian is the angle subtended by an arc of a circle that has the same length as the circle's radius (k = 1 in the formula given earlier). One full circle is 2π radians, and one radian is 180/π degrees, or about 57.2958 degrees. The radian is abbreviated rad, though this symbol is often omitted in mathematical texts, where radians are assumed unless specified otherwise. The radian is used in virtually all mathematical work beyond simple practical geometry, due, for example, to the pleasing and "natural" properties that the trigonometric functions display when their arguments are in radians. The radian is the (derived) unit of angular measurement in the SI system.

• The mil is approximately equal to a milliradian. There are several definitions.

• The full circle (or revolution, rotation, full turn or cycle) is one complete revolution. The revolution and rotation are abbreviated rev and rot, respectively, but just r in rpm (revolutions per minute). 1 full circle = 360° = 2π rad = 400 gon = 4 right angles.

• The right angle is 1/4 of a full circle. It is the unit used in Euclid's Elements. 1 right angle = 90° = π/2 rad = 100 gon.

• The angle of the equilateral triangle is 1/6 of a full circle. It was the unit used by the Babylonians, 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 = 60° = π/3 rad ≈ 1.047197551 rad.
• The grad, also called grade, gradian, or gon is 1/400 of a full circle, so one full circle is 400 grads and a right angle is 100 grads. It is a decimal subunit of the right angle. A kilometer was historically defined as a centi-gon of arc along a great circle of the Earth, so the kilometer is the decimal analog to the sexagesimal nautical mile. The gon is used mostly in triangulation.

• The point, used in navigation, is 1/32 of a full circle. It is a binary subunit of the full circle. Naming all 32 points on a compass rose is called "boxing the compass". 1 point = 1/8 of a right angle = 11.25° = 12.5 gon.

• The astronomical hour angle is 1/24 of a full circle. The sexagesimal subunits were called minute of time and second of time (even though they are units of angle). 1 hour = 15° = π/12 rad = 1/6 right angle ≈ 16.667 gon.

• The binary degree, also known as the binary radian (or brad), is 1/256 of a full circle. The binary degree is used in computing so that an angle can be efficiently represented in a single byte.

• The grade of a slope, or gradient, is not truly an angle measure (unless it is explicitly given in degrees, as is occasionally the case). Instead it is equal to the tangent of the angle, or sometimes the sine. Gradients are often expressed as a percentage. For the usual small values encountered (less than 5%), the grade of a slope is approximately the measure of an angle in radians.

This article describes the unit of angle. ... The sexagesimal (base-sixty) is a numeral system with sixty as the base. ... For other uses, see Astronomy (disambiguation). ... For other uses, see Ballistics (disambiguation). ... This article does not cite any references or sources. ... A nautical mile or sea mile is a unit of length. ... For the Brisbane bus routes known collectively as the Great Circle Line (598 & 599), see the following list of Brisbane Transport routes A great circle on a sphere A great circle is a circle on the surface of a sphere that has the same diameter as the sphere, dividing the... A second of arc or arcsecond is a unit of angular measurement which comprises one-sixtieth of an arcminute, or 1/3600 of a degree of arc or 1/1296000 ≈ 7. ... Image File history File links Angle_radian. ... Image File history File links Angle_radian. ... For the musical group, see Radian (band). ... Sine redirects here. ... Look up si, Si, SI in Wiktionary, the free dictionary. ... The mil (in full, angular mil) is a unit of angular measure. ... The radian is a unit of plane angle. ... ... For other uses, see Revolutions per minute (disambiguation). ... This article is about angles in geometry. ... The frontispiece of Sir Henry Billingsleys first English version of Euclids Elements, 1570 Euclids Elements (Greek: ) is a mathematical and geometric treatise consisting of 13 books written by the Greek mathematician Euclid in Alexandria circa 300 BC. It comprises a collection of definitions, postulates (axioms), propositions (theorems... For alternate meanings, such as the musical instrument, see triangle (disambiguation). ... Babylonia was a state in southern Mesopotamia, in modern Iraq, combining the territories of Sumer and Akkad. ... The sexagesimal (base-sixty) is a numeral system with sixty as the base. ... The grad is a measurement of plane angles of value 1/400 of a full circle, thus dividing a right angle in 100. ... This article is about angles in geometry. ... A kilometer (Commonwealth spelling: kilometre), symbol: km is a unit of length in the metric system equal to 1,000 metres (from the Greek words χίλια (khilia) = thousand and μέτρο (metro) = count/measure). ... Centi (symbol c) is a SI prefix in the SI system of units denoting a factor of 10-2, or 1/100. ... Triangulation can be used to find the distance from the shore to the ship. ... This article is about determination of position and direction on or above the surface of the earth. ... A common compass rose as is found on a nautical chart showing both true and magnetic north with magnetic declination A compass rose is a figure displaying the orientation of the cardinal directions, north, south, east and west on a map or nautical chart. ... A modern compass card. ... In astronomy, an objects hour angle (HA) is defined as the difference between the current local sidereal time (LST) and the right ascension () of the object: HAobject = LST - object Thus, the objects hour angle indicates how much sidereal time has passed since the object was on the local... For the computer industry magazine, see Byte (magazine). ... A grade (or gradient) is the pitch of a slope, and is often expressed as a percent tangent, or rise over run. It is used to express the steepness of slope on a hill, stream, roof, railroad, or road, where zero indicates level (with respect to gravity) and increasing numbers... In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ...

Positive and negative angles

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 many geometrical situations a negative angle of −θ is effectively equivalent to a positive angle of "one full rotation less θ". For example, a clockwise rotation of 45° (that is, an angle of −45°) is often effectively equivalent to a counterclockwise rotation of 360° − 45° (that is, an angle of 315°). A clockwise motion is one that proceeds like the clocks hands: from the top to the right, then down and then to the left, and back to the top. ... A clockwise motion is one that proceeds like the clocks hands: from the top to the right, then down and then to the left, and back to the top. ... Cartesian means relating to the French mathematician and philosopher Descartes, who, among other things, worked to merge algebra and Euclidean geometry. ... Cartesian means relating to the French mathematician and philosopher Descartes, who, among other things, worked to merge algebra and Euclidean geometry. ...

In three dimensional geometry, "clockwise" and "counterclockwise" 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. This article is about vectors that have a particular relation to the spatial coordinates. ...

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. This article is about determination of position and direction on or above the surface of the earth. ... In navigation, a bearing is the clockwise angle between a reference direction (or a datum line) and the direction to an object. ...

Approximations

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

Identifying angles

In mathematical expressions, it is common to use Greek letters (α, β, γ, θ, φ, ...) to serve as variables standing for the size of some angle. (To avoid confusion with its other meaning, the symbol π is not used for this purpose.) Lower case roman letters (a, b, c, ...) are also used. See the figures in this article for examples. Technical note: Due to technical limitations, some web browsers may not display some special characters in this article. ... In computer science and mathematics, a variable (pronounced ) (sometimes called an object or identifier in computer science) is a symbolic representation used to denote a quantity or expression. ... When a circles diameter is 1, its circumference is Ï€. Pi or Ï€ is the ratio of a circles circumference to its diameter in Euclidean geometry, approximately 3. ...

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 ∠BAC or BÂC. 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, ∠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 ∠BAC always refers to the anticlockwise (positive) angle from B to C, and ∠CAB to the anticlockwise (positive) angle from C to B.

Types of angles

Right angle.

Acute (a), obtuse (b), and straight (c) angles. Here, a and b are supplementary angles.

Reflex angle.

The complementary angles a and b (b is the complement of a, and a is the complement of b).

• An angle of 90° (π/2 radians, or one-quarter of the full circle) is called a right angle.

  Two lines that form a right angle are said to be perpendicular or orthogonal.

• Angles smaller than a right angle (less than 90°) are called acute angles ("acute" meaning "sharp").
• Angles larger than a right angle and smaller than two right angles (between 90° and 180°) are called obtuse angles ("obtuse" meaning "blunt").
• Angles equal to two right angles (180°) are called straight angles.
• Angles larger than two right angles but less than a full circle (between 180° and 360°) are called reflex angles.
• Angles that have the same measure are said to be 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. These angles are congruent.
• 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 (90°) 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 (180°) are called supplementary angles.

  The difference between an angle and a straight angle is termed the supplement of the angle.

• Two angles that sum to one full circle (360°) are called explementary angles or conjugate angles.
• An angle that is part of a simple polygon is called an interior angle if it lies in the inside of that the simple polygon. Note that in a simple polygon that is concave, at least one interior angle exceeds 180°.

  In Euclidean geometry, the measures of the interior angles of a triangle add up to π radians, or 180°; the measures of the interior angles of a simple quadrilateral add up to 2π radians, or 360°. In general, the measures of the interior angles of a simple polygon with n sides add up to [(n − 2) × π] radians, or [(n − 2) × 180]°.

• The angle supplementary to the interior angle is called the exterior angle. It measures the amount of "turn" one has to make at this vertex to trace out the polygon. If the corresponding interior angle exceeds 180°, 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 360°, one full turn.

• Some authors use the name exterior angle of a simple polygon to simply mean the explementary (not supplementary!) of the interior angle [1]. 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 (alternate) angles at the two points of intersection are congruent; adjacent angles are supplementary (that is, their measures add to π radians, or 180°).

Image File history File links Right_angle. ... Image File history File links Right_angle. ... Image File history File links Angle_obtuse_acute_straight. ... Image File history File links Angle_obtuse_acute_straight. ... Look up supplementary in Wiktionary, the free dictionary. ... Image File history File links Reflex_angle. ... Image File history File links Reflex_angle. ... Image File history File links Complement_angle. ... Image File history File links Complement_angle. ... A pair of complementary angles, because they add up to 90 degrees. ... When a circles diameter is 1, its circumference is π. Pi or π is the ratio of a circles circumference to its diameter in Euclidean geometry, approximately 3. ... This article is about angles in geometry. ... Fig. ... In mathematics, orthogonal is synonymous with perpendicular when used as a simple adjective that is not part of any longer phrase with a standard definition. ... As an abstract term, congruence means similarity between objects. ... Two lines intersect to create two pairs of vertical angles. ... In the illustration, angles A and B are adjacent. ... A pair of complementary angles, because they both add up to the sum of 90 degrees. ... Look up supplementary in Wiktionary, the free dictionary. ... A simple concave hexagon In geometry, two edges of a polygon may cross or even overlap in general. ... In geometry, an internal angle is an angle that 2 sides of a polygon form by touching. ... Euclid Euclidean geometry is a mathematical system attributed to the Greek mathematician [[Euclid]] of Alexandria. ... For alternate meanings, such as the musical instrument, see triangle (disambiguation). ... This article is about the geometric shape. ... Look up polygon in Wiktionary, the free dictionary. ... Negative has meaning in several contexts: Look up negative in Wiktionary, the free dictionary. ... In mathematics, an orientation on a real vector space is a choice of which ordered bases are positively oriented (or right-handed) and which are negatively oriented (or left-handed). ... This article is about the mathematical construct. ... An open surface with X-, Y-, and Z-contours shown. ... This article is about the mathematical construct. ... For the game magazine, see Polyhedron (magazine). ... In Aerospace engineering, the dihedral is the angle that the two wings make with each other. ... A surface normal, or just normal to a flat surface is a three-dimensional vector which is perpendicular to that surface. ... Transversal t cuts two parallel lines, a and b. ... Parallel is a term in geometry and in everyday life that refers to a property in Euclidean space of two or more lines or planes, or a combination of these. ... In the illustration, angles A and B are adjacent. ... Look up supplementary in Wiktionary, the free dictionary. ...

A formal definition

Using trigonometric functions

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

and

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 there correspond two angles in the geometric range 0 < θ < 2π, since

Using rotations

Suppose we have two unit vectors and in the euclidean plane . Then there exists one positive isometry (a rotation), and one only, from to that maps u onto v. Let r be such a rotation. Then the relation defined by is an equivalence relation and we call angle of the rotation r the equivalence class , where denotes the unit circle of . The angle between two vectors will simply be the angle of the rotation that maps one onto the other. We have no numerical way of determining an angle yet. To do this, we choose the vector (1,0), then for any point M on at distance θ from (1,0) (on the circle), let . If we call r_θ the rotation that transforms (1,0) into , then is a bijection, which means we can identify any angle with a number between 0 and 2π. In mathematics, an isometry, isometric isomorphism or congruence mapping is a distance-preserving isomorphism between metric spaces. ... In mathematics, given a set X and an equivalence relation ~ on X, the equivalence class of an element a in X is the subset of all elements in X which are equivalent to a: [a] = { x ∈ X | x ~ a } The notion of equivalence classes is useful for constructing sets out...

Angles between curves

The angle between the two curves 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:—amphicyrtic (Gr. ἀμφί, on both sides, κυρτόσ, convex) or cissoidal (Gr. κισσόσ, ivy), biconvex; xystroidal or sistroidal (Gr. ξυστρίσ, a tool for scraping), concavo-convex; amphicoelic (Gr. κοίλη, a hollow) or angulus lunularis, biconcave. Image File history File links Curve_angles. ... Image File history File links Curve_angles. ... In mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object. ... For other uses, see tangent (disambiguation). ...

The dot product and generalisation

In the Euclidean plane, the angle θ between two vectors u and v is related to their dot product and their lengths by the formula Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called Euclidean geometry, which is the study of the relationships between angles and distances in space. ... This article is about vectors that have a particular relation to the spatial coordinates. ... In mathematics, the dot product, also known as the scalar product, is a binary operation which takes two vectors over the real numbers R and returns a real-valued scalar quantity. ...

This allows one to define angles in any real inner product space, replacing the Euclidean dot product · by the Hilbert space inner product <·,·>. In mathematics, an inner product space is a vector space with additional structure, an inner product (also called a scalar product), which allows us to introduce geometrical notions such as angles and lengths of vectors. ... The mathematical concept of a Hilbert space (named after the German mathematician David Hilbert) generalizes the notion of Euclidean space in a way that extends methods of vector algebra from the plane and three-dimensional space to spaces of functions. ...

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 g_ij are the components of the metric tensor G, In differential geometry, Riemannian geometry is the study of smooth manifolds with Riemannian metrics, i. ... In mathematics, in Riemannian geometry, the metric tensor is a tensor of rank 2 that is used to measure distance and angle in a space. ... For other uses, see tangent (disambiguation). ...

Angles in geography and astronomy

In geography we specify the location of any point on the Earth 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. Map of Earth showing lines of latitude (horizontally) and longitude (vertically), Eckert VI projection; large version (pdf, 1. ... This article is about the geographical term. ... Longitude is the east-west geographic coordinate measurement most commonly utilized in cartography and global navigation. ... World map showing the equator in red In tourist areas, the equator is often marked on the sides of roads The equator marked as it crosses Ilhéu das Rolas, in São Tomé and Príncipe. ... The Prime Meridian, Greenwich The Prime Meridian is the meridian (line of longitude) passing through the Royal Greenwich Observatory, Greenwich, England; it is the meridian at which longitude is 0 degrees. ...

In astronomy, we similarly specify a given point on the celestial sphere using any of several Astronomical coordinate systems, where the references vary according to the particular system. For other uses, see Astronomy (disambiguation). ... The celestial sphere is divided by the celestial equator. ... Astronomical coordinate systems are coordinate systems used in astronomy to describe the location of objects in the sky and in the universe. ...

Astronomers can also 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. This article is about the astronomical object. ... This article is about Earth as a planet. ...

Astronomers also measure the apparent size of objects. For example, the full moon has an angular measurement of approximately 0.5°, 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. Angular size is a measurement of how large or small something is using rotational measurement (degrees of arc, arc_minutes, and arc-seconds). ... For other uses, see Full Moon. ... The small-angle formula is a mathematical approximation, used in astronomy. ...

References

Notes

On-line resources

• Slocum, Johnathan. (2007) Preliminary Indo-European lexicon - Pokorny PIE data. Retrieved on 16 July 2007, from University of Texas research department: linguistics research center

is the 197th day of the year (198th in leap years) in the Gregorian calendar. ... Year 2007 (MMVII) is the current year, a common year starting on Monday of the Gregorian calendar and the AD/CE era in the 21st century. ...

See also

• Complementary angles
• Supplementary angles
• Central angle
• Inscribed angle
• Solid angle for a concept of angle in three dimensions.
• Astrological aspect
• Protractor
• Clock angle problem

A pair of complementary angles, because they add up to 90 degrees. ... Look up supplementary in Wiktionary, the free dictionary. ... Angle AOB forms a central angle of circle O A central angle is an angle whose vertex is the center of a circle, and whose sides pass through a pair of points on the circle, thereby subtending an arc between those two points whose angle is (by definition) equal to... In geometry, an inscribed angle is formed when two secant lines of a circle (or, in a degenerate case, when one secant line and one tangent line of that circle) intersect on the circle. ... A solid angle is the three dimensional analog of the ordinary angle. ... The astrological aspects are noted in the central circle of this natal chart, where the different colors and symbols distinguish between the different aspects, such as the square (red) or trine (blue) In astrology, an aspect is the relative angle between two heavenly bodies. ... The term protractor is used both in technics and surgery. ... The diagram shows the angles formed by the hands of an analog clock showing a time of 2:20 Clock angle problems are a type of mathematical problem which involve finding the angles between the hands of an analog clock. ...

External links

• Angle Bisectors in a Quadrilateral at cut-the-knot
• Constructing a triangle from its angle bisectors at cut-the-knot
• Convert angles in sexagesimal degree format to decimal degrees, and vice-versa
• Angle Estimation -- for basic astronomy.
• Angle definition pages with interactive applets.
• Various angle constructions with compass and straightedge Animated demonstrations

cut-the-knot is an educational website maintained by Alexander Bogomolny and devoted to popular exposition of a great variety of topics in mathematics. ... cut-the-knot is an educational website maintained by Alexander Bogomolny and devoted to popular exposition of a great variety of topics in mathematics. ... For other uses, see Astronomy (disambiguation). ...