Angle
In geometry, an angle is the figure formed by two rays, called the sides of the angle, sharing a common endpoint, called the vertex.[1] This configuration describes the amount of rotation or inclination between the two rays, which originates from ancient mathematical traditions and remains central to Euclidean geometry.[2] Angles are measured using units such as degrees or radians to quantify their size.[3] In the degree system, a full rotation around the vertex measures 360 degrees, while in radians, it measures $2\pi radians, providing a dimensionless measure based on the ratio of arc length to radius in a circle.[3] These units facilitate precise calculations in fields like trigonometry and navigation.[1] Based on their measure, angles are classified into several types: an acute angle measures less than 90 degrees, a right angle measures exactly 90 degrees, an obtuse angle measures greater than 90 degrees but less than 180 degrees, a straight angle measures 180 degrees, and a reflex angle measures greater than 180 degrees but less than 360 degrees.[4] These classifications are essential for understanding geometric relationships, such as those in triangles where the sum of interior angles is always 180 degrees.[5] The concept of an angle has evolved historically, with early definitions appearing in Euclid's Elements around 300 BCE, describing it as the inclination of two lines in a plane that meet without forming a straight line.[2] Ancient Egyptians and Babylonians applied practical geometric techniques in surveying and astronomy, contributing to the development of Greek mathematics, including the formalization of angles.[6] In modern mathematics, angles extend beyond plane geometry to vector spaces and inner products, enabling applications in physics, computer graphics, and engineering.[1]Definition and Fundamentals
Definition
In geometry, a ray is defined as a half-line that originates at a fixed point, known as the endpoint, and extends infinitely in one direction.[7] An angle is the geometric figure formed by two such rays, referred to as the sides or arms of the angle, that share a common endpoint called the vertex.[8] One ray is designated as the initial side, and the other as the terminal side; the measure of the angle corresponds to the amount of rotation about the vertex required to align the initial side with the terminal side.[8] This configuration lies within a plane, defining a plane angle as the inclination between the two rays that intersect but do not form a straight line. Plane angles are fundamental to two-dimensional geometry, whereas solid angles extend this concept to three dimensions as a measure of the cone-like region subtended by a surface at a point, equivalent to the area projected onto a unit sphere centered at that point.[9] The basic measure of a plane angle, denoted θ, is given by the ratio of the arc length s between the initial and terminal sides on a circle centered at the vertex to the radius r of that circle, specifically considering the smaller arc for angles less than or equal to 180°: \theta = \frac{s}{r} This formulation arises from the geometric properties of circular arcs and provides a dimensionless quantity for the angle's magnitude.[8] Basic classifications of plane angles based on their measures include the right angle, which measures exactly 90°; the acute angle, measuring less than 90°; and the obtuse angle, measuring between 90° and 180°.[10] For instance, the corner of a square forms a right angle, while angles in an equilateral triangle are acute. These examples illustrate how angle measures determine the spatial relationships in simple geometric figures.Notation and Representation
In geometry, angles are denoted using the symbol ∠ followed by three capital letters, where the middle letter specifies the vertex and the outer letters indicate the endpoints of the rays forming the angle, such as ∠ABC for the angle at B between points A, B, and C.[11][12] This three-letter convention ensures unambiguous identification, particularly in diagrams with multiple angles sharing a vertex.[12] For general or variable angles in mathematical expressions, Greek letters such as θ (theta), φ (phi), or α (alpha) are standardly employed to represent unknown or parametric angles.[13] In triangle geometry, angles are conventionally labeled with capital letters corresponding to their vertices—∠A, ∠B, ∠C—where ∠A lies opposite side a, ∠B opposite side b, and ∠C opposite side c.[14] Directed angles, which account for orientation (typically measured counterclockwise), may be represented using the variant symbol ∡, as in ∡ABC to denote the signed angle from ray BA to ray BC; arrows on rays in diagrams can further indicate direction.[15] In contrast, undirected angles emphasize magnitude alone and use the standard ∠ symbol without directional markers. Diagrammatic conventions often include a small arc drawn between the rays to visually delineate the angle, aiding clarity in sketches.[11] In typesetting systems like LaTeX, the command\angle generates the ∠ symbol for precise mathematical notation, such as $\angle ABC$.[16] To maintain precision in multifaceted figures, the full three-point notation is preferred over single-letter abbreviations, minimizing potential misinterpretation.[12]
Measurement and Units
Angles are measured by quantifying the amount of rotation between two rays sharing a common vertex, or equivalently, in the context of circular geometry, as the ratio of the arc length subtended by the angle to the radius of the circle.[17] For the radian unit, this principle is formalized as \theta = \frac{s}{r}, where \theta is the angle in radians, s is the arc length, and r is the radius.[18] The primary units for measuring angles are degrees and radians. A degree, denoted by the symbol °, divides a full circle into 360 equal parts, so one degree represents \frac{1}{360} of a complete rotation.[3] This system originates from the ancient Babylonian sexagesimal (base-60) numeral system, which facilitated subdivisions into minutes (1/60 of a degree) and seconds (1/60 of a minute) for precise measurements.[19] In contrast, the radian measure, often denoted simply as rad or without a unit, defines a full circle as $2\pi radians, where one radian corresponds to the angle subtended by an arc equal in length to the radius.[20] Other units include gradians (also called gons, denoted gon or grad), which divide a full circle into 400 equal parts for applications in surveying and engineering, and revolutions (or turns), where one revolution equals one complete rotation around a circle.[21][1] Conversions between units are essential for calculations across contexts. Specifically, \pi radians equals 180 degrees, leading to the general conversion formula \theta_{\text{rad}} = \theta_{\text{deg}} \times \frac{\pi}{180}.[22] From a dimensional analysis perspective, angles are fundamentally dimensionless quantities because they arise as ratios of lengths (arc length to radius), yielding a pure number independent of the unit system.[23] However, in some physical and engineering contexts, angles are assigned a distinct dimension [angle] to track consistency in equations involving trigonometric functions or rotations.[24] Common tools for measuring angles in degrees include the protractor, a semicircular instrument marked in degree increments, historically used for navigation and drafting since the early modern period.[25]Basic Properties and Operations
Types of Angles
Angles are classified based on their measures or relationships to other angles in plane geometry. Measure-based classifications categorize angles according to their degree values relative to a full circle of 360° or a straight line of 180°. An acute angle measures less than 90°[1]. A right angle measures exactly 90°[1]. An obtuse angle measures greater than 90° but less than 180°[1]. A straight angle measures exactly 180°[1]. A reflex angle measures greater than 180° but less than 360°[1]. Relationship-based classifications describe angles in terms of their positions and interactions with adjacent or intersecting elements. Adjacent angles share a common vertex and one common ray but do not overlap in their interiors[26]. Vertical angles are formed when two lines intersect, consisting of the pairs of opposite angles at the intersection point; these pairs are always equal in measure[1]. Complementary angles are two angles whose measures sum to 90°[1]. Supplementary angles are two angles whose measures sum to 180°[27]. The equality of vertical angles is established by the vertical angle theorem, a fundamental result in Euclidean geometry attributed to Thales of Miletus and formalized in Euclid's Elements (Book I, Proposition 15)[28]. The theorem states that if two straight lines intersect, the vertical angles formed are equal. The proof relies on the axioms of equality and the property that adjacent angles on a straight line sum to two right angles (180°). Consider two lines AB and CD intersecting at point E. Angles ∠AEC and ∠AED are adjacent on line CD and thus sum to 180°. Similarly, ∠AED and ∠DEB are adjacent on line AB and sum to 180°. By the transitivity of equality, subtracting the common ∠AED from both sums yields ∠AEC = ∠DEB. The same reasoning applies to the other pair of vertical angles, ∠AED = ∠BEC[28]. In examples involving intersecting lines, adjacent angles form linear pairs that are supplementary, summing to 180°, while the opposite vertical angles remain equal regardless of the specific measures. For instance, if one angle in the intersection measures 70°, its adjacent angle measures 110° (supplementary), and the opposite vertical angle also measures 70°[1]. These classifications apply to angles in the plane without reference to polygonal interiors.Angle Addition, Subtraction, and Equivalence
Angle addition refers to the operation of combining two angles to form a new angle, interpreted geometrically as the total rotation from the initial side of the first angle through the second. In Euclidean geometry, when two angles are adjacent—sharing a common vertex and a common side—the angle addition postulate asserts that the measure of the combined angle equals the sum of the measures of the individual angles. Formally, if point D lies in the interior of \angle BAC, then m\angle BAC = m\angle BAD + m\angle DAC, where m denotes the measure in degrees or radians.[29] This postulate underpins many constructions, such as determining unknown angles in diagrams where adjacent angles form a straight line or intersect. For instance, adding a 30° angle to a 60° angle yields a 90° right angle, illustrating how addition operationalizes angular relationships in polygons and figures.[29] Angle subtraction, the inverse of addition, measures the angular difference as a rotation in the opposite direction from one angle to another. Derived from the angle addition postulate, the angle subtraction theorem states that if \angle BAD \cong \angle B'A'D' and \angle BAC \cong \angle B'A'C' with D interior to \angle BAC and D' interior to \angle B'A'C', then \angle DAC \cong \angle D'A'C', allowing the isolation of the difference angle.[30] This operation is essential for applications involving directed angles, such as calculating deviations in geometric configurations or differences within triangles; for example, subtracting 60° from a 90° angle in a right triangle leaves a 30° remainder, aiding in side-length computations via the law of sines.[31] Angles exhibit equivalence through the concept of coterminal angles, which are angles that, when drawn in standard position, terminate at the same ray on the unit circle despite differing measures. Two angles \theta and \phi are coterminal if their difference is an integer multiple of a full rotation, specifically \theta - \phi = 360^\circ k for some integer k, or equivalently in radians, \theta - \phi = 2\pi k.[32] The general formula for generating coterminal angles is \theta' = \theta + 360^\circ n, \quad n \in \mathbb{Z}, where adding or subtracting multiples of $360^\circ (or $2\pi radians) accounts for complete revolutions without altering the terminal side. This equivalence arises from the periodic nature of the circle: a proof follows from the fact that rotating by $360^\circ returns any ray to its original position, so iterative additions preserve the endpoint on the circumference.[33] Coterminal angles thus represent the same orientation modulo full rotations, facilitating consistent trigonometric evaluations across equivalent measures.[22]Signed and Reference Angles
In trigonometry, angles can be assigned a sign to indicate direction of rotation from the initial side. A signed angle is positive if measured counterclockwise from the positive x-axis and negative if measured clockwise. This convention arises in the standard position, where the vertex of the angle is at the origin and the initial side lies along the positive x-axis; the terminal side then determines the angle's measure.[34][35] The reference angle provides a way to simplify calculations by relating any angle to an acute angle with the same trigonometric properties relative to the axes. Defined as the acute angle formed between the terminal side of the given angle (in standard position) and the nearest x-axis, the reference angle is always between 0° and 90° (or 0 and π/2 radians). Its value depends on the quadrant: in Quadrant I, it equals the angle θ itself; in Quadrant II, it is 180° - θ; in Quadrant III, θ - 180°; and in Quadrant IV, 360° - θ.[36][37][38] Reference angles are particularly useful for evaluating trigonometric functions of angles in any quadrant, as the functions' values can be determined from the reference angle with appropriate sign adjustments based on the quadrant. For instance, for an angle θ = 150° in Quadrant II, the reference angle is 30°, and sin(150°) = sin(30°) = 1/2, while cos(150°) = -cos(30°) = -√3/2. Similarly, a signed angle of 120° is coterminal with -240°, both sharing the same reference angle of 60° for trigonometric computations.[39][40] In the context of complex numbers, the argument arg(z) of a nonzero complex number z = x + iy represents the signed angle that the vector from the origin to the point (x, y) makes with the positive real axis, measured counterclockwise as positive. The principal argument is typically taken in the interval (-π, π], allowing negative values for points in the lower half-plane. This signed angle facilitates polar representation, z = |z| (cos θ + i sin θ), where θ = arg(z).[41][42][43]Angles in Euclidean Geometry
Angles Between Lines and Curves
In Euclidean geometry, the angle between two intersecting lines is defined as the smaller of the two angles formed at their point of intersection, which is always between 0° and 90°. This measure captures the deviation in direction between the lines. For lines in the coordinate plane with slopes m_1 and m_2, the tangent of this angle \phi is given by the formula \tan \phi = \left| \frac{m_2 - m_1}{1 + m_1 m_2} \right|. This formula derives from the difference in the inclinations of the lines, where the slope m = \tan \alpha represents the tangent of the angle \alpha that the line makes with the positive x-axis.[44] Special cases arise based on the relationship between the slopes. If m_1 = m_2, the numerator is zero, so \tan \phi = 0, implying \phi = 0^\circ; the lines are parallel and do not intersect unless coincident. If $1 + m_1 m_2 = 0 (or m_1 m_2 = -1), the denominator is zero, making \tan \phi undefined and thus \phi = 90^\circ; the lines are perpendicular. These conditions highlight fundamental orthogonality and parallelism in line configurations.[44] In coordinate geometry, lines are often represented using direction vectors to describe their orientation. A line with slope m has a direction vector \langle 1, m \rangle, which points along the line and encodes its steepness relative to the axes. This vector representation facilitates geometric analysis without relying on explicit intersection points. For example, consider the lines y = x (slope m_1 = 1) and y = -x (slope m_2 = -1). Substituting into the formula yields \tan \phi = \left| \frac{-1 - 1}{1 + (1)(-1)} \right| = \left| \frac{-2}{0} \right| = \infty, so \phi = 90^\circ, confirming the lines are perpendicular as expected from their symmetric orientations.[44] The concept extends naturally to curves in the plane. The angle between two curves at a point of intersection is the angle between their tangent lines at that point, determined by the slopes of the tangents. These slopes are obtained from the first derivatives: if the curves are given by y = f(x) and y = g(x), then m_1 = f'(x_0) and m_2 = g'(x_0) at the intersection (x_0, y_0), and the same formula for \tan \phi applies. This approach relies on local linear approximations via calculus to quantify the curves' directional difference. To compute it, first solve for intersection points by setting f(x) = g(x), then evaluate the derivatives there.[45]Interior and Exterior Angles in Polygons
In polygons, the interior angles are the angles formed at each vertex inside the closed shape. For a simple polygon with n sides, the sum of the interior angles is (n-2) \times 180^\circ. This formula arises from triangulating the polygon, which divides it into n-2 non-overlapping triangles; since each triangle has interior angles summing to $180^\circ, the total sum for the polygon is (n-2) \times 180^\circ. For example, a triangle (n=3) has interior angles summing to $180^\circ, a quadrilateral (n=4) to $360^\circ, and a pentagon (n=5) to $540^\circ. In a regular polygon, where all sides and angles are equal, each interior angle measures \frac{(n-2) \times 180^\circ}{n}. This follows directly from dividing the total interior angle sum by n. Irregular polygons also obey the same total sum formula, but individual angles vary; in convex polygons, all interior angles are less than $180^\circ, while in concave polygons, at least one interior angle exceeds $180^\circ. Exterior angles are formed by extending one side of the polygon at each vertex and measuring the angle between that extension and the adjacent side. The exterior angle at a vertex equals $180^\circ minus the interior angle at that same vertex, as they form a linear pair. For any simple convex polygon, the sum of the exterior angles, taken in one direction around the polygon, is always $360^\circ, corresponding to a full turn. In regular polygons, each exterior angle is \frac{360^\circ}{n}.Angle Bisectors and Trisectors
An angle bisector is a ray or line segment that divides an angle into two congruent angles of equal measure. In the context of a triangle, the internal angle bisector from a vertex intersects the opposite side and divides it into two segments proportional to the lengths of the adjacent sides, according to the angle bisector theorem: if the bisector from vertex A meets side BC at D, then \frac{BD}{DC} = \frac{AB}{AC}. This theorem holds for any triangle and provides a key property for solving geometric problems involving proportionality. The construction of an angle bisector using only a compass and straightedge is a classical Euclidean method. To bisect \angle ABC with vertex B, place the compass at B and draw an arc intersecting rays BA and BC at points P and Q, respectively. Then, from P and Q, draw equal arcs that intersect at a point R inside the angle. The line from B through R is the bisector. In an isosceles triangle ABC with AB = AC, the angle bisector from vertex A coincides with the median and altitude to base BC, simplifying constructions and proofs due to the symmetry. In coordinate geometry, the direction of the angle bisector between two rays originating from the origin with direction vectors \vec{u} and \vec{v} (neither zero) can be determined vectorially. The bisector direction is along the vector \frac{\vec{u}}{||\vec{u}||} + \frac{\vec{v}}{||\vec{v}||}, which normalizes the vectors to unit length before adding them, ensuring the result bisects the angle by equalizing the angular deviation. This formula arises from the property that the bisector equidistant in angular terms from the two rays corresponds to the sum of their unit directions. Angle trisectors divide an angle into three equal parts, but unlike bisection, arbitrary trisection cannot be achieved with compass and straightedge alone. Pierre Wantzel proved this impossibility in 1837 using field theory, showing that trisecting a general angle like $60^\circ requires constructing lengths not obtainable in quadratic extensions of the rationals, as the minimal polynomial for \cos(20^\circ) is cubic. However, exact trisection is possible with additional tools, such as the Archimedean spiral: draw the spiral r = \theta from the vertex, intersect it with a circle of appropriate radius centered at the vertex, and connect the intersection points to the vertex to form the trisectors. For practical purposes, approximate trisectors can be constructed using iterative methods, such as repeatedly halving the angle until sufficiently small and adjusting, or employing geometric approximations like D'Ocagne's method involving a semicircle and midpoints to achieve errors less than $0.1^\circ for typical angles. These approximations are useful in applications where exactness is not required, such as drafting or numerical simulations.Angles in Trigonometry and Circles
Circular Measurement
In circular geometry, the radian serves as the natural unit for measuring angles, defined as the central angle subtended by an arc whose length equals the radius of the circle.[46] Formally, the radian measure \theta is given by \theta = s / r, where s is the length of the arc and r is the radius.[46] Since the circumference of a circle is $2\pi r, a full rotation around the circle corresponds to an angle of $2\pi radians.[46] This definition ties angular measure directly to the geometry of the circle, making radians particularly suited for applications involving circular motion and curvature. The radian offers key advantages over other units, primarily because it is dimensionless, allowing seamless integration into physical and mathematical equations without unit conversion complications.[47] In calculus, radians simplify the derivatives of trigonometric functions; for example, the derivative of \sin [\theta](/page/Theta) is \cos [\theta](/page/Theta) only when \theta is in radians.[48] This property extends to rotational dynamics, where angular velocity \omega = d[\theta](/page/Theta) / dt naturally yields linear velocity v = r \omega in consistent units.[47] To relate radians to degrees, multiply the degree measure by \pi / 180; thus, $180^\circ = \pi radians.[49] Here, \pi \approx 3.14159. For small angles in radians, the approximation \sin \theta \approx \theta holds, which is valuable in approximations for pendulums and optics.[50] A practical consequence of the radian definition is the arc length formula s = r \theta, enabling direct computation of arc measures from angular subtends.[51] In a circle, the central angle \theta subtended by an arc of length s is \theta = s / r, representing the angle at the circle's center.[46] By contrast, an inscribed angle—formed by two chords sharing a common endpoint on the circumference and intercepting the same arc—measures half the central angle.[52] For instance, if a central angle is $2\pi / 3 radians (120°), the inscribed angle intercepting the same arc is \pi / 3 radians (60°).[52]Common Angles and Their Trigonometric Values
Common angles in trigonometry refer to specific measures, such as 0°, 30°, 45°, 60°, and 90°, along with their radian equivalents of 0, π/6, π/4, π/3, and π/2, whose exact trigonometric values are derived from fundamental geometric properties and are essential for simplifying calculations and proving identities.[53] These values are obtained primarily from the side ratios in special right triangles, such as the 30°-60°-90° triangle with sides in the ratio 1 : √3 : 2 and the 45°-45°-90° triangle with sides 1 : 1 : √2, applying the Pythagorean theorem to ensure exactness.[53] For instance, in the 30°-60°-90° triangle, the sine of 30° is the opposite side over the hypotenuse, yielding sin(30°) = 1/2, while cos(60°) = 1/2 follows similarly.[53] The unit circle further confirms these values, where the coordinates of points corresponding to these angles on the circle of radius 1 give (cos θ, sin θ); for example, at 45° or π/4, the point (√2/2, √2/2) provides both sin(π/4) = √2/2 and cos(π/4) = √2/2.[53] Tangent values are then computed as the ratio sin θ / cos θ, such as tan(60°) = √3.[54] The following table summarizes the exact values for sine, cosine, and tangent in the first quadrant:| Angle (degrees) | Angle (radians) | sin θ | cos θ | tan θ |
|---|---|---|---|---|
| 0° | 0 | 0 | 1 | 0 |
| 30° | π/6 | 1/2 | √3/2 | √3/3 |
| 45° | π/4 | √2/2 | √2/2 | 1 |
| 60° | π/3 | √3/2 | 1/2 | √3 |
| 90° | π/2 | 1 | 0 | undefined |