"pair of compasses. The ratio of the length s of the arc by the radius r of the circle is the measure of the angle in radians. The measure of the angle in another angular unit is then obtained by multiplying its measure in radians by the scaling factor , where ""k"" is the measure of a complete turn in the chosen unit (for example 360 for degrees or 400 for gradians): 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"
"study of dynamical systems. Unit circle In mathematics, a unit circle is a circle with a radius of one. Frequently, especially in trigonometry, the unit circle is the circle of radius one centered at the origin (0, 0) in the Cartesian coordinate system in the Euclidean plane. The unit circle is often denoted ; the generalization to higher dimensions is the unit sphere. If is a point on the unit circle's circumference, then and are the lengths of the legs of a right triangle whose hypotenuse has length 1. Thus, by the Pythagorean theorem, and satisfy the equation Since for"
"Unit circle In mathematics, a unit circle is a circle with a radius of one. Frequently, especially in trigonometry, the unit circle is the circle of radius one centered at the origin (0, 0) in the Cartesian coordinate system in the Euclidean plane. The unit circle is often denoted ; the generalization to higher dimensions is the unit sphere. If is a point on the unit circle's circumference, then and are the lengths of the legs of a right triangle whose hypotenuse has length 1. Thus, by the Pythagorean theorem, and satisfy the equation Since for all , and since"
"Radian The radian (SI symbol rad) is the SI unit for measuring angles, and is the standard unit of angular measure used in many areas of mathematics. The length of an arc of a unit circle is numerically equal to the measurement in radians of the angle that it subtends; one radian is just under 57.3 degrees (expansion at ). The unit was formerly an SI supplementary unit, but this category was abolished in 1995 and the radian is now considered an SI derived unit. Separately, the SI unit of solid angle measurement is the steradian. The radian is most"
"defined similarly; for example, the cosine of the angle is the ratio between the adjacent side and the hypotenuse, while the tangent gives the ratio between the opposite and adjacent sides. As stated, the value formula_2 appears to depend on the choice of right triangle containing an angle of measure ""α"". However, this is not the case: all such triangles are similar, and so the ratio is the same for each of them. In trigonometry, a unit circle is the circle of radius one centered at the origin (0, 0) in the Cartesian coordinate system. Let a line through the"
"Degree (angle) A degree (in full, a degree of arc, arc degree, or arcdegree), usually denoted by ° (the degree symbol), is a measurement of a plane angle, defined so that a full rotation is 360 degrees. It is not an SI unit, as the SI unit of angular measure is the radian, but it is mentioned in the SI brochure as an accepted unit. Because a full rotation equals 2 radians, one degree is equivalent to radians. The original motivation for choosing the degree as a unit of rotations and angles is unknown. One theory states that it is"
"kilometers of distance at the equator; 1 centigrad of arc equaled 1 kilometer; 0.1 cc (centi-centigrads) of arc equaled 1 meter. The Gradian is sometimes used in surveying in some parts of the world. Subdivisions of the gradian used in surveying can be referred to as c (short for centigrad) and cc (effectively centi-centigrad), where 1 c = 0.01 grad and 1 cc = 0.0001 grad. Gradian The gradian is a unit of measurement of an angle, equivalent to formula_1 of a turn, formula_2 of a degree, or formula_3 of a radian. The gradian is defined as formula_4 of the"
"the same proportion, so the ratio ""s""/""r"" is unaltered. (Proof. The formula above can be rewritten as One turn, for which units, corresponds to an arc equal in length to the circle's circumference, which is 2""r"", so . Substituting ""n"" for ""θ"" and 2""r"" for ""s"" in the formula, results in ) The angle addition postulate states that if B is in the interior of angle AOC, then formula_2 The measure of the angle AOC is the sum of the measure of angle AOB and the measure of angle BOC. In this postulate it does not matter in which unit"
"expressed in cylindrical coordinates is Arc lengths are denoted by ""s"", since the Latin word for length (or size) is ""spatium"". In the following lines, formula_69 represents the radius of a circle, formula_70 is its diameter, formula_71 is its circumference, formula_72 is the length of an arc of the circle, and formula_58 is the angle which the arc subtends at the centre of the circle. The distances formula_74 and formula_72 are expressed in the same units. Two units of length, the nautical mile and the metre (or kilometre), were originally defined so the lengths of arcs of great circles on"
"for example, the spherical angle formed by two great circles on a sphere is the dihedral angle between the planes determined by the great circles. ""Angle"" is also used to designate the measure of an angle or of a rotation. This measure is the ratio of the length of a circular arc to its radius. In the case of a geometric angle, the arc is centered at the vertex and delimited by the sides. In the case of a rotation, the arc is centered at the center of the rotation and delimited by any other point and its image by"
"geometrically by the following observation. In the first definition, the curvature of a circle is equal to the ratio of the angle of an arc to its length. Likewise, the curvature of a plane curve at any point is the limiting ratio of , an infinitesimal angle (in radians) between tangents to that curve at the ends of an infinitesimal segment of the curve, to the length of that segment , i.e., . If the tangents at the ends of the segment are represented by unit vectors, it is easy to show that in this limit, the magnitude of the"
"Steradian The steradian (symbol: sr) or square radian is the SI unit of solid angle. It is used in three-dimensional geometry, and is analogous to the radian, which quantifies planar angles. Whereas an angle in radians, projected onto a circle, gives a ""length"" on the circumference, a solid angle in steradians, projected onto a sphere, gives an ""area"" on the surface. The name is derived from the Greek στερεός ""stereos"" 'solid' + radian. The steradian, like the radian, is a dimensionless unit, the area subtended and the square of its distance from the center: both the numerator and denominator of"
"Minute and second of arc A minute of arc, arcminute (arcmin), arc minute, or minute arc is a unit of angular measurement equal to of one degree. Since one degree is of a turn (or complete rotation), one minute of arc is of a turn – it is for this reason that the Earth's circumference is almost exactly 21,600 nautical miles. A minute of arc is of a radian. A second of arc, arcsecond (arcsec), or arc second is of an arcminute, of a degree, of a turn, and (about ) of a radian. These units originated in Babylonian astronomy"
"for degrees or 400 for gradians): 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. (Proof. The formula above can be rewritten as One turn, for which units, corresponds to an arc equal in length to the circle's circumference, which is 2""r"", so . Substituting ""n"" for ""θ"" and 2""r"" for ""s"" in the formula, results in ) The angle addition postulate states that if ""B"" is in the interior of angle"
"etc. This convention is used throughout the article. When measuring in radians, an angle of θ radians will correspond to an arc whose length is rθ, where r is the radius of the circle. Thus, in the unit circle, ""the arc whose cosine is x"" is the same as ""the angle whose cosine is x"", because the length of the arc of the circle in radii is the same as the measurement of the angle in radians. Similarly, in computer programming languages the inverse trigonometric functions are usually called asin, acos, atan. The notations , , , etc., as introduced"
"of cosines states: Since this is a unit sphere, the lengths , and are simply equal to the angles (in radians) subtended by those sides from the center of the sphere. (For a non-unit sphere, the lengths are the subtended angles times the radius, and the formula still holds if and are reinterpreted as the subtended angles). As a special case, for , then , and one obtains the spherical analogue of the Pythagorean theorem: If the law of cosines is used to solve for , the necessity of inverting the cosine magnifies rounding errors when is small. In this"
"Angular unit Throughout history, angles have been measured in many different units. The most contemporary units are the degree ( ° ) and radian (rad), but many others have been used throughout history. The purpose of this page is to aggregate other concepts pertaining to the angular unit, where additional explanation can be provided. The size of a geometric angle is usually characterized by the magnitude of the smallest rotation that maps one of the rays into the other. Angles that have the same size are said to be ""equal"" or ""congruent"" or ""equal in measure"". In some contexts, such"
"unit circle: For a vertical chord ""AB"" of the unit circle, the sine of the angle ""θ"" (representing half of the subtended angle ""Δ"") is the distance ""AC"" (half of the chord). On the other hand, the versed sine of ""θ"" is the distance ""CD"" from the center of the chord to the center of the arc. Thus, the sum of cos(""θ"") (equal to the length of line ""OC"") and versin(""θ"") (equal to the length of line ""CD"") is the radius ""OD"" (with length 1). Illustrated this way, the sine is vertical (""rectus"", literally ""straight"") while the versine is horizontal"
"(μrad) and nanoradians (nrad) are used in astronomy, and can also be used to measure the beam quality of lasers with ultra-low divergence. More common is arc second, which is rad (around 4.8481 microradians). Similarly, the prefixes smaller than milli- are potentially useful in measuring extremely small angles. Radian The radian (SI symbol rad) is the SI unit for measuring angles, and is the standard unit of angular measure used in many areas of mathematics. The length of an arc of a unit circle is numerically equal to the measurement in radians of the angle that it subtends; one radian"
"Sagitta (geometry) In geometry, the sagitta (sometimes abbreviated as sag) of a circular arc is the distance from the center of the arc to the center of its base. It is used extensively in architecture when calculating the arc necessary to span a certain height and distance and also in optics where it is used to find the depth of a spherical mirror or lens. The name comes directly from Latin ""sagitta"", meaning an arrow. In the following equations, denotes the sagitta (the depth of the arc), equals the radius of the circle, and half the length of the chord"
"arcminutes and arcseconds to measure angles when the object being measured is too small for direct visual inspection. For instance, a toolmaker's optical comparator will often include an option to measure in ""minutes and seconds"". Minute and second of arc A minute of arc, arcminute (arcmin), arc minute, or minute arc is a unit of angular measurement equal to of one degree. Since one degree is of a turn (or complete rotation), one minute of arc is of a turn – it is for this reason that the Earth's circumference is almost exactly 21,600 nautical miles. A minute of arc"
"Measurement of a Circle Measurement of a Circle or Dimension of the Circle (Greek: , ""Kuklou metrēsis"") is a treatise that consists of three propositions by Archimedes, ca. 250 BCE. The treatise is only a fraction of what was a longer work. Proposition one states: The area of any circle is equal to a right-angled triangle in which one of the sides about the right angle is equal to the radius, and the other to the circumference, of the circle. Any circle with a circumference ""c"" and a radius ""r"" is equal in area with a right triangle with the"
"Degree of curvature Degree of curve or degree of curvature is a measure of curvature of a circular arc used in civil engineering for its easy use in layout surveying. The degree of curvature is defined as the central angle to the ends of an arc or chord of agreed length. Various lengths are commonly used in different areas of practice. This angle is also the change in forward direction as that portion of the curve is traveled. Curvature is usually measured in radius of curvature. A small circle can be easily laid out by just using radius of curvature,"
"all points (""x"", ""y"") such that This equation, known as the Equation of the Circle, follows from the Pythagorean theorem applied to any point on the circle: as shown in the adjacent diagram, the radius is the hypotenuse of a right-angled triangle whose other sides are of length |""x"" − ""a""| and |""y"" − ""b""|. If the circle is centred at the origin (0, 0), then the equation simplifies to Parametric form <br> The equation can be written in parametric form using the trigonometric functions sine and cosine as where ""t"" is a parametric variable in the range 0 to"
"is thus a right triangle, and is similar to C′CA since they share the angle at C′. Thus all three corresponding sides are in the same proportion; in particular, we have C′A : C′C = C′P : C′A and AP : C′A = CA : C′C. The center of the circle, O, bisects A′A, so we also have triangle OAP similar to A′AB, with OP half the length of A′B. In terms of side lengths, this gives us In the first equation C′P is C′O+OP, length ""r""+⁄""c"", and C′C is the diameter, 2""r"". For a unit circle we have the"
"be labeled φ = π/2 − θ) has cotangent equal to the length of the other leg, and cosecant equal to the length of the hypotenuse. In that way, this trigonometric identity involving the cotangent and the cosecant also follows from the Pythagorean theorem. The following table gives the identities with the factor or divisor that relates them to the main identity. The unit circle centered at the origin in the Euclidean plane is defined by the equation: Given an angle θ, there is a unique point ""P"" on the unit circle at an angle θ from the ""x""-axis, and"
"this system. In most mathematical work beyond practical geometry, angles are typically measured in radians rather than degrees. This is for a variety of reasons; for example, the trigonometric functions have simpler and more ""natural"" properties when their arguments are expressed in radians. These considerations outweigh the convenient divisibility of the number 360. One complete turn (360°) is equal to 2"""" radians, so 180° is equal to radians, or equivalently, the degree is a mathematical constant: 1° = . The turn (or revolution, full circle, full rotation, cycle) is used in technology and science. One turn is equal to 360°."
"Area of a circle In geometry, the area enclosed by a circle of radius is . Here the Greek letter represents a constant, approximately equal to 3.14159, which is equal to the ratio of the circumference of any circle to its diameter. One method of deriving this formula, which originated with Archimedes, involves viewing the circle as the limit of a sequence of regular polygons. The area of a regular polygon is half its perimeter multiplied by the distance from its center to its sides, and the corresponding formula (that the area is half the perimeter times the radius, i.e."
"Semicircle In mathematics (and more specifically geometry), a semicircle is a one-dimensional locus of points that forms half of a circle. The full arc of a semicircle always measures 180° (equivalently, radians, or a half-turn). It has only one line of symmetry (reflection symmetry). In non-technical usage, the term ""semicircle"" is sometimes used to refer to a half-disk, which is a two-dimensional geometric shape that also includes the diameter segment from one end of the arc to the other as well as all the interior points. By Thales' theorem, any triangle inscribed in a semicircle with a vertex at each"
"Gradian The gradian is a unit of measurement of an angle, equivalent to formula_1 of a turn, formula_2 of a degree, or formula_3 of a radian. The gradian is defined as formula_4 of the right angle (in other words, there are 100 gradians in the right angle), which implies a full turn being 400 gradians. It is also known as gon (from Greek /""gōnía"" for angle), grad, or grade. In continental Europe, the French term ""centigrade"" was in use for one hundredth of a grad. This was one reason for the adoption of the term Celsius to replace centigrade as"
"Inscribed angle In geometry, an inscribed angle is the angle formed in the interior of a circle when two secant lines (or, in a degenerate case, when one secant line and one tangent line of that circle) intersect on the circle. It can also be defined as the angle subtended at a point on the circle by two given points on the circle. Equivalently, an inscribed angle is defined by two chords of the circle sharing an endpoint. The inscribed angle theorem relates the measure of an inscribed angle to that of the central angle subtending the same arc. The"
"angle may be expressed in different units: Throughout history, carpenters and masons have known a quick way to confirm if an angle is a true ""right angle."" It is based on the most widely known Pythagorean triple (3, 4, 5) and so called the ""Rule of 3-4-5."" From the angle in question, running a straight line along one side exactly three units in length, and along the second side exactly four units in length, will create a hypotenuse (the longer line opposite the right angle that connects the two measured endpoints) of exactly 5 units in length. This measurement can"
"Circumference In geometry, the circumference (from Latin ""circumferentia"", meaning ""carrying around"") of a circle is the (linear) distance around it. That is, the circumference would be the length of the circle if it were opened up and straightened out to a line segment. Since a circle is the edge (boundary) of a disk, circumference is a special case of perimeter. The perimeter is the length around any closed figure and is the term used for most figures excepting the circle and some circular-like figures such as ellipses. Informally, ""circumference"" may also refer to the edge itself rather than to the"