"a formula for the surface area, we can use the same kind of “onion” approach we used for the disk. 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"
"letter alone to represent the ratio of a circle's circumference to its diameter was by Welsh mathematician William Jones in his 1706 work ""Synopsis Palmariorum Matheseos; or, a New Introduction to the Mathematics"". The Greek letter first appears there in the phrase ""1/2 Periphery ()"" in the discussion of a circle with radius one. However, he writes that his equations for are from the ""ready pen of the truly ingenious Mr. John Machin"", leading to speculation that Machin may have employed the Greek letter before Jones. Jones' notation was not immediately adopted by other mathematicians, with the fraction notation still"
"of a circle's circumference to its diameter, and because it is the first Greek letter in the name of the ancient sculptor Phidias. Although Martin Gardner later wrote that Phidias was chosen because he was ""believed to have used the golden proportion frequently in his sculpture"", Barr himself denied this, writing in his paper ""Parameters of beauty"" that he doubted Phidias used the golden ratio. Schooling communicated some of his discoveries with Barr to Cook after seeing an article by Cook about phyllotaxis, the arrangement of leaves on a plant stem, which often approximates the golden ratio. Schooling published his"
"is used to indicate the diameter of a circular section; for example, ""⌀14"" means the diameter of the circle is 14 units. In Unicode, there are multiple forms of the phi letter: In ordinary Greek text, the character U+03C6 φ is used exclusively, although this character has considerable glyphic variation, sometimes represented with a glyph more like the representative glyph shown for U+03C6 (φ, the ""loopy"" or ""open"" form) and less often with a glyph more like the representative glyph shown for U+03D5 (ϕ, the ""stroked"" or ""closed"" form). Unicode makes an effort to distinguish the two by generally calling"
"Circumflex The circumflex is a diacritic in the Latin and Greek scripts that is used in the written forms of many languages and in various romanization and transcription schemes. It received its English name from Latin ""circumflexus"" ""bent around""a translation of the Greek περισπωμένη (""perispōménē""). The circumflex in the Latin script is chevron-shaped ( ˆ ), while the Greek circumflex may be displayed either like a tilde ( ˜ ) or like an inverted breve ( ̑ ). In English the circumflex, like other diacritics, is sometimes retained on loanwords that used it in the original language (for example, ""crème"
"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"
"Omicron Omicron (uppercase Ο, lowercase ο, literally ""small o"": όμικρον < ὂ μικρόν - ""ò mikrón"", micron meaning 'small' in contrast to ""omega"") is the 15th letter of the Greek alphabet. In the system of Greek numerals it has a value of 70. This letter is derived from the Phoenician letter ayin . In classical Greek, omicron represented the sound in contrast to ""omega"" and ""ου"" . In modern Greek, omicron represents the . Letters that arose from omicron include Roman O and Cyrillic O. The upper-case letter of omicron (O) was originally used in mathematics as a symbol for"
"being used as late as 1767. Euler started using the single-letter form beginning with his 1727 ""Essay Explaining The Properties Of Air"", though he used , the ratio of radius to periphery, in this and some later writing. Euler first used in his 1736 work ""Mechanica"", and continued in his widely-read 1748 work ""Introductio in analysin infinitorum"" (he wrote: ""for the sake of brevity we will write this number as ; thus is equal to half the circumference of a circle of radius 1""). Because Euler corresponded heavily with other mathematicians in Europe, the use of the Greek letter spread"
"center of the ellipse. For example, conjugate diameters have the property that a tangent line to the ellipse at the endpoint of one of them is parallel to the other one. The longest diameter is called the major axis. The word ""diameter"" is derived from Greek διάμετρος (""diametros""), ""diameter of a circle"", from διά (""dia""), ""across, through"" and μέτρον (""metron""), ""measure"". It is often abbreviated DIA, dia, d, or ⌀. The definitions given above are only valid for circles, spheres and convex shapes. However, they are special cases of a more general definition that is valid for any kind of"
"Radius In classical geometry, a radius of a circle or sphere is any of the line segments from its center to its perimeter, and in more modern usage, it is also their length. The name comes from the Latin ""radius"", meaning ray but also the spoke of a chariot wheel. The plural of radius can be either ""radii"" (from the Latin plural) or the conventional English plural ""radiuses"". The typical abbreviation and mathematical variable name for radius is r. By extension, the diameter d is defined as twice the radius: If an object does not have a center, the term"
"as well as a few Aegean islands, namely Paros, Thasos and Melos, chose the exact opposite innovation, using a broken-up circle for the short and a closed circle for the long /o/. The name Ωμέγα is Byzantine; in Classical Greek, the letter was called ""ō"" (), whereas the omicron was called ""ou"" (). The modern lowercase shape goes back to the uncial form , a form that developed during the 3rd century BC in ancient handwriting on papyrus, from a flattened-out form of the letter () that had its edges curved even further upward. In addition to the Greek alphabet,"
"successive golden ratio series. Martin Ohm is believed to have been the first to use the term ""goldener Schnitt"" (golden section) to describe the ratio in 1835. In the century, Röber theorized that the Great Pyramid of Giza was based on a Kepler triangle. By 1914, mathematician Mark Barr suggested Greek letter Phi (φ) as a symbol for the golden ratio. Previously, it was represented by τ (tau, the first letter of the ancient Greek root τομή—meaning ""cut"" or ""section""). In 1966, ""Fibonacci Quarterly"" published φ calculated to 4,600 digits; thirty years later, it was calculated to decimal places. In"
"symbol "". is commonly defined as the ratio of a circle's circumference to its diameter : The ratio is constant, regardless of the circle's size. For example, if a circle has twice the diameter of another circle it will also have twice the circumference, preserving the ratio . This definition of implicitly makes use of flat (Euclidean) geometry; although the notion of a circle can be extended to any curved (non-Euclidean) geometry, these new circles will no longer satisfy the formula . Here, the circumference of a circle is the arc length around the perimeter of the circle, a quantity"
"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"
"formatting to indicate the style of text. Omega Omega (capital: Ω, lowercase: ω; Greek ὦ, later ὦ μέγα, Modern Greek ωμέγα) is the 24th and last letter of the Greek alphabet. In the Greek numeric system/Isopsephy (Gematria), it has a value of 800. The word literally means ""great O"" (""ō mega"", mega meaning ""great""), as opposed to omicron, which means ""little O"" (""o mikron"", micron meaning ""little""). In phonetic terms, the Ancient Greek Ω is a long open-mid ""o"" , comparable to the vowel of British English ""raw"". In Modern Greek, Ω represents the , the same sound as omicron."
"be produced by an iterative geometrical construction suggested by Archimedes' Stomachion in the setting of the regular dodecagon. In this case, the task is to give rational approximations to the tangent of π/12. 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"
"the normal Greek letters, with markup and formatting to indicate text style. Sigma Sigma (uppercase Σ, lowercase σ, lowercase in word-final position ς; ) is the eighteenth letter of the Greek alphabet. In the system of Greek numerals, it has a value of 200. When used at the end of a word (when the word is not all caps), the final form (ς) is used, e.g. (Odysseus); note the two sigmas in the center of the name, and the word-final sigma at the end. The shape and alphabetic position of sigma is derived from Phoenician shin 𐤔 . The name"
"Circles of Apollonius The circles of Apollonius are any of several sets of circles associated with Apollonius of Perga, a renowned Greek geometer. Most of these circles are found in planar Euclidean geometry, but analogs have been defined on other surfaces; for example, counterparts on the surface of a sphere can be defined through stereographic projection. The main uses of this term are fivefold: A circle is usually defined as the set of points P at a given distance ""r"" (the circle's radius) from a given point (the circle's center). However, there are other, equivalent definitions of a circle. Apollonius"
"have where again formula_12 is the length of the semi-major axis and formula_13 is the eccentricity formula_14 In graph theory the circumference of a graph refers to the longest (simple) cycle contained in that graph. 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"
"letter Ef (Ф, ф) descends from phi. Phi is also used as a symbol for the golden ratio and on other occasions in math and science. This use is separately encoded as the Unicode glyph ϕ. The modern Greek pronunciation of the letter is sometimes encountered in English (as ) when the letter is being used in this sense. The lowercase letter φ (or often its variant, ϕ) is often used to represent the following: The uppercase letter Φ is used as a symbol for: The diameter symbol in engineering, ⌀, is often erroneously referred to as ""phi"". This symbol"
"of Cnidus in the fifth century B.C. had found that the area of a disk is proportional to its radius squared. Archimedes used the tools of Euclidean geometry to show that the area inside a circle is equal to that of a right triangle whose base has the length of the circle's circumference and whose height equals the circle's radius in his book ""Measurement of a Circle"". The circumference is 2""r"", and the area of a triangle is half the base times the height, yielding the area ""r"" for the disk. Prior to Archimedes, Hippocrates of Chios was the first"
"from the empty set symbol ∅, from an (italic) uppercase phi ""Φ"", and from the Nordic vowel Ø. See also slashed zero. In German, the diameter symbol (German """") is also used as an average symbol (""Durchschnittszeichen""). Diameter In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints lie on the circle. It can also be defined as the longest chord of the circle. Both definitions are also valid for the diameter of a sphere. In more modern usage, the length of a diameter is also called"
"Lune of Hippocrates In geometry, the lune of Hippocrates, named after Hippocrates of Chios, is a lune bounded by arcs of two circles, the smaller of which has as its diameter a chord spanning a right angle on the larger circle. Equivalently, it is a non-convex plane region bounded by one 180-degree circular arc and one 90-degree circular arc. It is the first curved figure to have its exact area calculated mathematically. Hippocrates wanted to solve the classic problem of squaring the circle, i.e. constructing a square by means of straightedge and compass, having the same area as a given"
"in his book ""Measurement of a Circle"". (The circumference is 2""r"", and the area of a triangle is half the base times the height, yielding the area ""r"" for the disk.) Archimedes approximated the value of π (and hence the area of a unit-radius circle) with his doubling method, in which he inscribed a regular triangle in a circle and noted its area, then doubled the number of sides to give a regular hexagon, then repeatedly doubled the number of sides as the polygon's area got closer and closer to that of the circle (and did the same with circumscribed"
"Omega Omega (capital: Ω, lowercase: ω; Greek ὦ, later ὦ μέγα, Modern Greek ωμέγα) is the 24th and last letter of the Greek alphabet. In the Greek numeric system/Isopsephy (Gematria), it has a value of 800. The word literally means ""great O"" (""ō mega"", mega meaning ""great""), as opposed to omicron, which means ""little O"" (""o mikron"", micron meaning ""little""). In phonetic terms, the Ancient Greek Ω is a long open-mid ""o"" , comparable to the vowel of British English ""raw"". In Modern Greek, Ω represents the , the same sound as omicron. The letter omega is transcribed ""ō"" or"
"number. The use of the Greek letter formula_1 to denote the ratio of a circle's circumference to its diameter was also popularized by Euler (although it did not originate with him). He is also credited for inventing the notation ""i"" to denote formula_2. Euler made important contributions to complex analysis. He introduced the scientific notation. He discovered what is now known as Euler's formula, that for any real number formula_3, the complex exponential function satisfies This has been called ""the most remarkable formula in mathematics "" by Richard Feynman. Euler's identity is a special case of this: This identity is"
"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. 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 is typically not used for this purpose.) Lower case Roman letters (""a"", ""b"", ""c"", . . . ) are also used,"
"to its diameter and its radius. That is to say, there exists a constant number pi, (the Greek ""p"" for perimeter), such that if is the circle's perimeter and its diameter then, In terms of the radius of the circle, this formula becomes, To calculate a circle's perimeter, knowledge of its radius or diameter and the number suffices. The problem is that is not rational (it cannot be expressed as the quotient of two integers), nor is it algebraic (it is not a root of a polynomial equation with rational coefficients). So, obtaining an accurate approximation of is important in"
"an approximate method, though less accurate, documented in the ""Shulba Sutras"". Archimedes proved the formula for the area of a circle (, where is the radius of the circle) and showed that the value of lay between 3 + 1/7 (approximately 3.1429) and 3 + 10/71 (approximately 3.1408). See Numerical approximations of π for more on the history. The first known Greek to be associated with the problem was Anaxagoras, who worked on it while in prison. Hippocrates of Chios squared certain lunes, in the hope that it would lead to a solution — see Lune of Hippocrates. Antiphon the"
"simply ""o"". As the last letter of the Greek alphabet, Omega is often used to denote the last, the end, or the ultimate limit of a set, in contrast to alpha, the first letter of the Greek alphabet. Ω was not part of the early (8th century BC) Greek alphabets. It was introduced in the late 7th century BC in the Ionian cities of Asia Minor to denote the long half-open [ɔː]. It is a variant of omicron (Ο), broken up at the side (), with the edges subsequently turned outward (, , , ). The Dorian city of Knidos"
"Sigma Sigma (uppercase Σ, lowercase σ, lowercase in word-final position ς; ) is the eighteenth letter of the Greek alphabet. In the system of Greek numerals, it has a value of 200. When used at the end of a word (when the word is not all caps), the final form (ς) is used, e.g. (Odysseus); note the two sigmas in the center of the name, and the word-final sigma at the end. The shape and alphabetic position of sigma is derived from Phoenician shin 𐤔 . The name of sigma, according to one hypothesis, may continue that of Phoenician Samekh."
"the ratio of circumference to diameter, following earlier abbreviations for the Greek word ""periphery"" (περιφέρεια) by Oughtred and others. His 1711 work ""Analysis per quantitatum series, fluxiones ac differentias"" introduced the dot notation for differentiation in calculus. In 1731 he published ""Discourses of the Natural Philosophy of the Elements"". He was noticed and befriended by two of Britain’s foremost mathematicians – Edmund Halley and Sir Isaac Newton – and was elected a fellow of the Royal Society in 1711. He later became the editor and publisher of many of Newton’s manuscripts and built up an extraordinary library that was one"
"The equivalent terms used in codicology, manuscript studies, print studies and publishing are ""recto"" and ""verso"". Generally, the side of a coin with the larger-scale image will be called the obverse (especially if the image is a single head) and, if that does not serve to distinguish them, the side that is more typical of a wide range of coins from that location will be called the obverse. Following this principle, in the most famous of ancient Greek coins, the tetradrachm of Athens, the obverse is the head of Athena and the reverse is her owl. Similar versions of these"