query: who discovered the formula for the circumference of a circle

pos: Circumference pi, is represented by the Greek letter . The first few decimal digits of the numerical value of are 3.141592653589793 ... Pi is defined as the ratio of a circle's circumference to its diameter : Or, equivalently, as the ratio of the circumference to twice the radius. The above formula can be rearranged to solve for the circumference: The use of the mathematical constant is ubiquitous in mathematics, engineering, and science. In "Measurement of a Circle" written circa 250 BCE, Archimedes showed that this ratio (, since he did not use the name ) was greater than 3 but less

pos: Indiana Pi Bill which he interpreted literally as finding a square with the same area as the circle. He knew that Archimedes' formula for the area of a circle, which calls for multiplying the diameter by one fourth of the circumference, is not considered a solution to the ancient problem of squaring the circle. This is because the problem is to "construct" the area using compass and straightedge only, and Archimedes did not give a method for constructing a straight line with the same length as the circumference. Apparently, Goodwin was unaware of this central requirement; he believed that the problem with the

pos: Area of a circle 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

pos: Circumference pi, is represented by the Greek letter . The first few decimal digits of the numerical value of are 3.141592653589793 ... Pi is defined as the ratio of a circle's circumference to its diameter : Or, equivalently, as the ratio of the circumference to twice the radius. The above formula can be rearranged to solve for the circumference: The use of the mathematical constant is ubiquitous in mathematics, engineering, and science. In "Measurement of a Circle" written circa 250 BCE, Archimedes showed that this ratio (, since he did not use the name ) was greater than 3 but less

pos: Area of a circle 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.

pos: Measurement of a Circle 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

pos: Circle calculus. Early science, particularly geometry and astrology and astronomy, was connected to the divine for most medieval scholars, and many believed that there was something intrinsically "divine" or "perfect" that could be found in circles. Some highlights in the history of the circle are: The ratio of a circle's circumference to its diameter is (pi), an irrational constant approximately equal to 3.141592654. Thus the length of the circumference "C" is related to the radius "r" and diameter "d" by: As proved by Archimedes, in his Measurement of a Circle, the area enclosed by a circle is equal to that of

pos: On Spirals on the spiral when it has completed one turn. Let the tangent at P cut the line perpendicular to OP at T. OT is the length of the circumference of the circle with radius OP. Archimedes had already proved as the first proposition of "Measurement of a Circle" that the area of a circle is equal to a right-angled triangle having the legs' lengths equal to the radius of the circle and the circumference of the circle. So the area of the circle with radius OP is equal to the area of the triangle OPT. On Spirals On Spirals ()

pos: Area of a circle good approximation to the circumference. To compute "u" and "U" for large "n", Archimedes derived the following doubling formulae: Starting from a hexagon, Archimedes doubled "n" four times to get a 96-gon, which gave him a good approximation to the circumference of the circle. In modern notation, we can reproduce his computation (and go further) as follows. For a unit circle, an inscribed hexagon has "u" = 6, and a circumscribed hexagon has "U" = 4. Doubling seven times yields The last entry of the table has ⁄ as one of its best rational approximations; i.e., there is no better

pos: Area of a circle 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

pos: Area 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

neg: Pi 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

neg: Area angles are arbitrarily close to 0° and the other two are arbitrarily close to 180°. For a circle, the ratio of the area to the circumference (the term for the perimeter of a circle) equals half the radius "r". This can be seen from the area formula "πr" and the circumference formula 2"πr". The area of a regular polygon is half its perimeter times the apothem (where the apothem is the distance from the center to the nearest point on any side). Doubling the edge lengths of a polygon multiplies its area by four, which is two (the ratio of

neg: Area of a circle trigonometry, uses the coarea formula. Define a function formula_10 by formula_11. Note ρ is a Lipschitz function whose gradient is a unit vector formula_12 (almost everywhere). Let "D" be the disc formula_13 in formula_14. We will show that formula_15, where formula_16 is the two-dimensional Lebesgue measure in formula_14. We shall assume that the one-dimensional Hausdorff measure of the circle formula_18 is formula_19, the circumference of the circle of radius "r". (This can be taken as the definition of circumference.) Then, by the coarea formula, Similar to the onion proof outlined above, we could exploit calculus in a different way in

neg: Law of sines analog of the formula in Euclidean geometry expressing the sine of an angle as the opposite side divided by the hypotenuse. Define a generalized sine function, depending also on a real parameter : The law of sines in constant curvature reads as By substituting , , and , one obtains respectively the Euclidean, spherical, and hyperbolic cases of the law of sines described above. Let indicate the circumference of a circle of radius in a space of constant curvature . Then . Therefore the law of sines can also be expressed as: This formulation was discovered by János Bolyai. For

neg: Area of a circle "area" and "circumference". The analytical definitions are seen to be equivalent, if it is agreed that the circumference of the circle is measured as a rectifiable curve by means of the integral The integral appearing on the right is an abelian integral whose value is a half-period of the sine function, equal to . Thus formula_6 is seen to be true as a theorem. Several of the arguments that follow use only concepts from elementary calculus to reproduce the formula formula_1, but in many cases to regard these as actual proofs, they rely implicitly on the fact that one can

neg: Haversine formula tend to arise in the formula when finite precision is used. Because is then large (approaching , half the circumference) a small error is often not a major concern in this unusual case (although there are other great-circle distance formulas that avoid this problem). (The formula above is sometimes written in terms of the arctangent function, but this suffers from similar numerical problems near = 1.) As described below, a similar formula can be written using cosines (sometimes called the spherical law of cosines, not to be confused with the law of cosines for plane geometry) instead of haversines, but

neg: Circumference 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

neg: Radius theory, the radius of a graph is the minimum over all vertices "u" of the maximum distance from "u" to any other vertex of the graph. The radius of the circle with perimeter (circumference) "C" is For many geometric figures, the radius has a well-defined relationship with other measures of the figure. The radius of a circle with area is The radius of the circle that passes through the three non-collinear points , , and is given by where is the angle . This formula uses the law of sines. If the three points are given by their coordinates ,

neg: Circle "p" = 2, giving the familiar In taxicab geometry, "p" = 1. Taxicab circles are squares with sides oriented at a 45° angle to the coordinate axes. While each side would have length formula_35 using a Euclidean metric, where "r" is the circle's radius, its length in taxicab geometry is 2"r". Thus, a circle's circumference is 8"r". Thus, the value of a geometric analog to formula_36 is 4 in this geometry. The formula for the unit circle in taxicab geometry is formula_37 in Cartesian coordinates and in polar coordinates. A circle of radius 1 (using this distance) is the von

neg: Taxicab geometry would have length formula_10 using a Euclidean metric, where "r" is the circle's radius, its length in taxicab geometry is 2"r". Thus, a circle's circumference is 8"r". Thus, the value of a geometric analog to formula_11 is 4 in this geometry. The formula for the unit circle in taxicab geometry is formula_12 in Cartesian coordinates and in polar coordinates. A circle of radius 1 (using this distance) is the von Neumann neighborhood of its center. A circle of radius "r" for the Chebyshev distance (L metric) on a plane is also a square with side length 2"r" parallel to the

neg: History of geometry Moscow Mathematical Papyrus gives the only ancient example finding the volume of a frustum of a pyramid, describing the correct formula: The Babylonians may have known the general rules for measuring areas and volumes. They measured the circumference of a circle as three times the diameter and the area as one-twelfth the square of the circumference, which would be correct if "π" is estimated as 3. The volume of a cylinder was taken as the product of the base and the height, however, the volume of the frustum of a cone or a square pyramid was incorrectly taken as the

neg: Bhaskara I's sine approximation formula using modern concepts and tools. Around half a dozen methods have been suggested, each based on a separate set of premises. Most of these derivations use only elementary concepts. Let the circumference of a circle be measured in degrees and let the radius "R" of the circle be also measured in degrees. Choosing a fixed diameter "AB" and an arbitrary point "P" on the circle and dropping the perpendicular "PM" to "AB", we can compute the area of the triangle "APB" in two ways. Equating the two expressions for the area one gets (1/2) "AB" × "PM" = (1/2) "AP"

neg: Area circle into sectors, as shown in the figure to the right. Each sector is approximately triangular in shape, and the sectors can be rearranged to form an approximate parallelogram. The height of this parallelogram is , and the width is half the circumference of the circle, or . Thus, the total area of the circle is : Though the dissection used in this formula is only approximate, the error becomes smaller and smaller as the circle is partitioned into more and more sectors. The limit of the areas of the approximate parallelograms is exactly , which is the area of

neg: Madhava series = "r"θ, "x" = "kotijya" = "r" cos θ and "y" = "jya" = "r" sin θ. Then "y" / "x" = tan θ. Substituting these in the last expression and simplifying we get Letting tan θ = "q" we finally have The second part of the quoted text specifies another formula for the computation of the circumference "c" of a circle having diameter "d". This is as follows. Since "c" = π "d" this can be reformulated as a formula to compute π as follows. This is obtained by substituting "q" = formula_22 (therefore "θ" = π / 6)

neg: Medallion knitting Medallion knitting Medallion knitting produces flat knitted fabrics that are circular or polygons using a technique similar to the crochet of doilies, by progressively increasing or decreasing the radius. The most difficult part of flat medallion knitting is increasing or decreasing at just the right rate. The circumference formula_1 of a circle is related to its radius formula_2 by the formula formula_3. Therefore, the number of stitches formula_4 in a row at radius formula_2 must exactly equal that circumference times the stitch gauge, if the medallion is to lie flat. If the number of stitches is "fewer" than this, the

neg: Angle 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

neg: Spirograph where formula_3 has been shrunk into a point. (Division by formula_78 in the formula is not a problem since both formula_63 and formula_85 are bounded functions). The other extreme case formula_79 corresponds to the inner circle formula_3's radius formula_88 matching the radius formula_2 of the outer circle formula_1, i.e. formula_91. In this case the trajectory is a single point. Intuitively, formula_3 is too large to roll inside the same-sized formula_1 without slipping. If formula_94 then the point formula_8 is on the circumference of formula_3. In this case the trajectories are called hypocycloids and the equations above reduce to those for

neg: Non-integer representation and the length of the string of symbols needed to express a given range of values. Base π can be used to more easily show the relationship between the diameter of a circle to its circumference, which corresponds to its perimeter; since circumference = diameter × π, a circle with a diameter 1 will have a circumference of 10, a circle with a diameter 10 will have a circumference of 100, etc. Furthermore, since the area = π × radius, a circle with a radius of 1 will have an area of 10, a circle with a radius of 10

neg: Circumference 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

neg: Iris (anatomy) in a circular motion, and a set of dilator muscles (dilator pupillae) which pull the iris radially to enlarge the pupil, pulling it in folds. The circle circumference sphincter constricting muscle is the opposing muscle of the circle-radius dilator muscle. The iris inner smaller circle-circumference changes size when constricting or dilating. The iris outer larger circle-circumference does not change size. The constricting muscle is located on the iris inner smaller circle-circumference. The back surface is covered by a heavily pigmented epithelial layer that is two cells thick (the iris pigment epithelium), but the front surface has no epithelium. This anterior

neg: Isoperimetric inequality area of a disk of radius "R" is "πR" and the circumference of the circle is 2"πR", so both sides of the inequality are equal to 4"π""R" in this case. Dozens of proofs of the isoperimetric inequality have been found. In 1902, Hurwitz published a short proof using the Fourier series that applies to arbitrary rectifiable curves (not assumed to be smooth). An elegant direct proof based on comparison of a smooth simple closed curve with an appropriate circle was given by E. Schmidt in 1938. It uses only the arc length formula, expression for the area of a plane

neg: Yigu yanduan square and circle respective ? Anwwer: The circumference of the square is 240 steps, the circumference of the circle is 60 steps." Method: set up tian yuan one (celetial element 1) as the diameter of the circle, x multiply it by 3 to get the circumference of the circle 3x (pi ~~3) subtract this from the sum of circumferences to obtain the circumference of the square formula_1 The square of it equals to 16 times the area of the square formula_2 Again set up tian yuan 1 as the diameter of circle, square it up and multiplied by 12 to

neg: Area of a circle to eight sides and so on. For a polygon with 2"n" sides, the parallelogram will have a base of length "ns", and a height "h". As the number of sides increases, the length of the parallelogram base approaches half the circle circumference, and its height approaches the circle radius. In the limit, the parallelogram becomes a rectangle with width "r" and height "r". There are various equivalent definitions of the constant π. The conventional definition in pre-calculus geometry is the ratio of the circumference of a circle to its diameter: However, because the circumference of a circle is not a

neg: Circumference length of the edge. The circumference of a circle is the distance around it, but if, as in many elementary treatments, distance is defined in terms of straight lines, this can not be used as a definition. Under these circumstances, the circumference of a circle may be defined as the limit of the perimeters of inscribed regular polygons as the number of sides increases without bound. The term circumference is used when measuring physical objects, as well as when considering abstract geometric forms. The circumference of a circle is related to one of the most important mathematical constants. This constant,

neg: Āryabhaṭa's sine table Āryabhaṭa has chosen the number 3438 as the value of radius of the base circle for the computation of his sine table. The rationale of the choice of this parameter is the idea of measuring the circumference of a circle in angle measures. In astronomical computations distances are measured in degrees, minutes, seconds, etc. In this measure, the circumference of a circle is 360° = (60 × 360) minutes = 21600 minutes. The radius of the circle, the measure of whose circumference is 21600 minutes, is 21600 / 2π minutes. Computing this using the value π = 3.1416 known to

neg: Aristotle's wheel paradox its movement from any point to another can be calculated by using an inverse of their ratio. The paradox is that the smaller inner circle moves 2πR, the circumference of the larger outer circle with radius R, rather than its own circumference. If the inner circle were rolled separately, it would move 2πr, its own circumference with radius r. The inner circle is not separate but rigidly connected to the larger. So 2πr is a red herring. If the smaller circle depends on the larger one (Case I), then the larger circle forces the smaller one to traverse the larger

neg: Hjelmslev transformation the case, line AH and EJ will be hyperparallel, and therefore will never meet. Consequently, any line drawn perpendicular to base AE between A and E must necessarily cross line AH at some finite distance. Johannes Hjelmslev discovered from this a method of compressing an entire hyperbolic plane into a finite circle. By applying this process to every line within the plane, one could compress this plane so that infinite spaces could be seen as planar. Hjelmslev's transformation would not yield a proper circle however. The circumference of the circle does not have a corresponding location within the plane, and

neg: Madhava of Sangamagrama and rational approximations of infinite series. However, as stated above, which results are precisely Madhava's and which are those of his successors, are somewhat difficult to determine. The following presents a summary of results that have been attributed to Madhava by various scholars. Among his many contributions, he discovered the infinite series for the trigonometric functions of sine, cosine, tangent and arctangent, and many methods for calculating the circumference of a circle. One of Madhava's series is known from the text "Yuktibhāṣā", which contains the derivation and proof of the power series for inverse tangent, discovered by Madhava. In the

neg: Penn–Calvert boundary dispute in Chestertown, Maryland. They subsequently met four times at New Castle's Court House, once in Joppa, Maryland, and once in Philadelphia. The Commission could not agree on several points of contention related to the Twelve Mile Circle. First, the Maryland Commissioners insisted that a circle must have a center point and they were not empowered to determine what the center point was. Second, the Maryland Commissioners insisted the circle should have a circumference of 12 miles, while the Pennsylvania Commissioners insisted it should have a radius of 12 miles. Lord Baltimore had also discovered the mapping error that resulted in

neg: Yuktibhāṣā three, "Kuttakaram", circle and disquisition on R-Sine; and the latter about astronomy. As per the old Indian tradition of starting off new chapters with elementary content, the first four chapters of the "Yuktibhasa" contain elementary mathematics, such as division, proof of Pythagorean theorem, square root determination, etc. The radical ideas are not discussed until the sixth chapter on circumference of a circle. Yuktibhasa contains the derivation and proof of the power series for inverse tangent, discovered by Madhava. In the text, Jyesthadeva describes Madhava's series in the following manner: This yields which further yields the theorem sometimes mistakenly attributed to

neg: Contributions of Leonhard Euler to mathematics 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

neg: Cognitive semantics Langacker, who makes a distinction between the notions of "profile" and "base". The profile is the concept symbolized by the word itself, while the base is the encyclopedic knowledge that the concept presupposes. For example, let the definition of "radius" be "a line segment that joins the center of a circle with any point on its circumference". If all we know of the concept "radius" is its profile, then we simply know that it is a line segment that is attached to something called the "circumference" in some greater whole called the "circle". That is to say, our understanding is

neg: Solid angle is much, much farther away. The solid angle subtended by the complete ()-dimensional spherical surface of the unit sphere in -dimensional Euclidean space can be defined in any number of dimensions . One often needs this solid angle factor in calculations with spherical symmetry. It is given by the formula where is the gamma function. When is an integer, the gamma function can be computed explicitly. It follows that This gives the expected results of 4 steradians for the 3D sphere bounded by a surface of area and 2 radians for the 2D circle bounded by a circumference of length

neg: Circumference than 3 by calculating the perimeters of an inscribed and a circumscribed regular polygon of 96 sides. This method for approximating was used for centuries, obtaining more accuracy by using polygons of larger and larger number of sides. The last such calculation was performed in 1630 by Christoph Grienberger who used polygons with 10 sides. Circumference is used by some authors to denote the perimeter of an ellipse. There is no general formula for the circumference of an ellipse in terms of the semi-major and semi-minor axes of the ellipse that uses only elementary functions. However, there are approximate formulas

neg: Arc (geometry) center of the circle, measure the angle where the two lines meet the center, then solve for L by cross-multiplying the statement: For example, if the measure of the angle is 60 degrees and the circumference is 24 inches, then This is so because the circumference of a circle and the degrees of a circle, of which there are always 360, are directly proportional. The area of the sector formed by an arc and the center of a circle (bounded by the arc and the two radii drawn to its endpoints) is The area "A" has the same proportion to

neg: Heron's formula Expressing Heron's formula with a Cayley–Menger determinant in terms of the squares of the distances between the three given vertices, illustrates its similarity to Tartaglia's formula for the volume of a three-simplex. Another generalization of Heron's formula to pentagons and hexagons inscribed in a circle was discovered by David P. Robbins. If , , , , , are lengths of edges of the tetrahedron (first three form a triangle; opposite to and so on), then where Heron's formula In geometry, Heron's formula (sometimes called Hero's formula), named after Hero of Alexandria, gives the area of a triangle when the length

neg: Earth's circumference determine latitude at sea. He reasoned that the lines of latitude could be used as the basis for a unit of measurement for distance and proposed the nautical mile as one minute or one-sixtieth () of one degree of latitude. As one degree is of a circle, one minute of arc is of a circle – such that the polar circumference of the Earth would be exactly 21,600 miles. Gunter used Snell's circumference to define a nautical mile as 6,080 feet, the length of one minute of arc at 48 degrees latitude. In 1791, the French Academy of Sciences selected

neg: Ellipse is formula_349) and stretch it by a factor formula_350 to make an ellipse. This scales the area by the same factor: formula_351 It is also easy to rigorously prove the area formula using integration as follows. Equation () can be rewritten as formula_352 For formula_353 this curve is the top half of the ellipse. So twice the integral of formula_354 over the interval formula_355 will be the area of the ellipse: The second integral is the area of a circle of radius formula_357 that is, formula_358 So An ellipse defined implicitly by formula_360 has area formula_361 The circumference formula_13 of

neg: Megagon sides of length "a" is given by The perimeter of a regular megagon inscribed in the unit circle is: which is very close to 2π. In fact, for a circle the size of the Earth's equator, with a circumference of 40,075 kilometres, one edge of a megagon inscribed in such a circle would be slightly over 40 meters long. The difference between the perimeter of the inscribed megagon and the circumference of this circle comes to less than 1/16 millimeters. Because 1000000 = 2 × 5, the number of sides is not a product of distinct Fermat primes and a

neg: Virasena effectively base-2 logarithms. He also worked with logarithms in base 3 (trakacheda) and base 4 (caturthacheda). Virasena gave the approximate formula "C" = 3"d" + (16"d"+16)/113 to relate the circumference of a circle, "C", to its diameter, "d". For large values of "d", this gives the approximation π ≈ 355/113 = 3.14159292..., which is more accurate than the approximation π ≈ 3.1416 given by Aryabhata in the "Aryabhatiya". Virasena was proficient in astrology, grammar, logic, mathematics and prosody. He wrote "Dhavala", a commentary on Jain canon "Shatakhandagama". He also started the work on "Jayadhavalaa" which was competed by his disciples.

neg: Chinese mathematics substitution. Problems in The Nine Chapters on the Mathematical Art take pi to be equal to three in calculating problems related to circles and spheres, such as spherical surface area. There is no explicit formula given within the text for the calculation of pi to be three, but it is used throughout the problems of both The Nine Chapters on the Mathematical Art and the Artificer's Record, which was produced in the same time period. Historians believe that this figure of pi was calculated using the 3:1 relationship between the circumference and diameter of a circle. Some Han mathematicians attempted

neg: Posidonius in the star's elevation indicated the distance between the two locales was 1/48 of the circle, he multiplied 5,000 by 48 to arrive at a figure of 240,000 stadia for the circumference of the earth. Translating stadia into modern units of distance can be problematic, but it is generally thought that the stadium used by Posidonius was almost exactly 1/10 of a modern statute mile. Thus Posidonius's measure of 240,000 stadia translates to , not much short of the actual circumference of . Posidonius was informed in his approach to finding the Earth's circumference by Eratosthenes, who a century earlier

neg: Four-dimensional space care must be exercised not to accept results that are not more rigorously tested. For example, consider the formulas for the circumference of a circle formula_11 and the surface area of a sphere: formula_12. One might be tempted to suppose that the surface volume of a hypersphere is formula_13, or perhaps formula_14, but either of these would be wrong. The correct formula is formula_15. Four-dimensional space A four-dimensional space or 4D space is a mathematical extension of the concept of three-dimensional or 3D space. Three-dimensional space is the simplest possible generalization of the observation that one only needs three numbers,

neg: Liu Hui's π algorithm by its radius, and the area doubles; hence multiply half the circumference by the radius to yield the area of circle"". When , half the circumference of the -gon approaches a semicircle, thus half a circumference of a circle multiplied by its radius equals the area of the circle. Liu Hui did not explain in detail this deduction. However it is self-evident by using Liu Hui's "in-out complement principle" which he provided elsewhere in "The Nine Chapters on the Mathematical Art": Cut up a geometric shape into parts, rearrange the parts to form another shape, the area of the two

neg: Cauchy's integral formula where formula_50 is the surface area of a unit ball in the space (that is, formula_51, the circumference of a circle with radius 1, and formula_52, the surface area of a sphere with radius 1). By definition of a Green's function, formula_53. It is this useful property that can be used, in conjunction with the generalized Stokes theorem: where, for an formula_55-dimensional vector space, formula_56 is an formula_57-vector and formula_58 is an formula_55-vector. The function formula_60 can, in principle, be composed of any combination of multivectors. The proof of Cauchy's integral theorem for higher dimensional spaces relies on the using

neg: Aristotle's wheel paradox circle’s circumference. If the larger circle depends on the smaller one (Case II), then the smaller circle forces the larger one to traverse the smaller circle’s circumference. This is the simplest solution. This solution considers the transition from starting to ending positions. Let Pb be a point on the bigger circle and Ps be a point on the smaller circle, both on the same radius. For convenience, assume they are both directly below the center, analogous to both hands of a clock pointing towards six. Both Pb and Ps travel a cycloid path as they roll together one revolution. The

neg: Aristotle's wheel paradox by both circles are the same length, as depicted by the blue and red dashed lines. The distance for the larger circle equals its circumference, but the distance for the smaller circle is longer than its circumference: a paradox or problem. The paradox is not limited to a wheel. Other things depicted in two dimensions show the same behavior. A roll of tape does. A typical round bottle rolled on its side does -- the smaller circle depicting the mouth or neck of the bottle. There are a few things that would be depicted with the brown horizontal line in

neg: Area to Heron's was discovered by the Chinese independently of the Greeks. It was published in 1247 in "Shushu Jiuzhang" ("Mathematical Treatise in Nine Sections"), written by Qin Jiushao. In the 7th century CE, Brahmagupta developed a formula, now known as Brahmagupta's formula, for the area of a cyclic quadrilateral (a quadrilateral inscribed in a circle) in terms of its sides. In 1842 the German mathematicians Carl Anton Bretschneider and Karl Georg Christian von Staudt independently found a formula, known as Bretschneider's formula, for the area of any quadrilateral. The development of Cartesian coordinates by René Descartes in the 17th century

neg: Antarctic Circle no permanent human population south of the Antarctic Circle, but there are several research stations in Antarctica operated by various nations that are inhabited by teams of scientists who rotate on a seasonal basis. In previous centuries some semi-permanent whaling stations were established on the continent, and some whalers would live there for a year or more. At least three children have been born in Antarctica, albeit in stations north of the Antarctic Circle. The circumference of the Antarctic Circle is roughly . The area south of the Circle is about and covers roughly 4% of Earth's surface. The continent

neg: Earth's circumference system of latitude and longitude. Longditude lines, also known as meridian lines, follow the polar circumference, and are all exactly the same length. Latitude lines, known as circles of latitude are parallel to each other; the equatorial circumference is the largest circle of latitude, and the Earth’s longest circumference distance. Other circles of latitude have smaller circumference; for example, the 60th parallel north or south is approximately half as long as the Equator. According to Cleomedes' "On the Circular Motions of the Celestial Bodies", around 240 BC, Eratosthenes, the librarian of the Library of Alexandria, calculated the circumference of the

neg: Ancient technology a 60-minute hour, which we still use today. They also divided up the circle into 360 degrees. They had a wide knowledge of mathematics including addition, subtraction, multiplication, division, quadratic and cubic equations, and fractions. This was important in keeping track of records as well as in some of their large building projects. The Mesopotamians had formulas for figuring out the circumference and area for different geometric shapes like rectangles, circles, and triangles. Some evidence suggests that they even knew the Pythagorean Theorem long before Pythagoras wrote it down. They may have even discovered the number for pi in figuring

neg: Bertrand–Diguet–Puiseux theorem Bertrand–Diguet–Puiseux theorem In the mathematical study of the differential geometry of surfaces, the Bertrand–Diguet–Puiseux theorem expresses the Gaussian curvature of a surface in terms of the circumference of a geodesic circle, or the area of a geodesic disc. The theorem is named for Joseph Bertrand, Victor Puiseux, and Charles François Diguet. Let "p" be a point on a smooth surface "M". The geodesic circle of radius "r" centered at "p" is the set of all points whose geodesic distance from "p" is equal to "r". Let "C"("r") denote the circumference of this circle, and "A"("r") denote the area of the

neg: History of trigonometry also gave the power series of π and the θ, radius, diameter, and circumference of a circle in terms of trigonometric functions. His works were expanded by his followers at the Kerala School up to the 16th century. The Indian text the Yuktibhāṣā contains proof for the expansion of the sine and cosine functions and the derivation and proof of the power series for inverse tangent, discovered by Madhava. The Yuktibhāṣā also contains rules for finding the sines and the cosines of the sum and difference of two angles. In China, Aryabhata's table of sines were translated into the Chinese

neg: Elizabeth Zimmermann Other patterns and techniques for which she is well known are the so-called "Pi Shawl," a circular shawl that Zimmermann claimed was formed by regularly spaced increases based on Pi -- as she said in her book Knitter's Almanac, "The geometry of the circle hing[es] on the mysterious relationship of the circumference of a circle to its radius. A circle will double its circumference in infinitely themselves-doubling distances, or, in knitters' terms, the distance between the increase-rounds, in which you double the number of stitches, goes 3, 6, 12, 24 and so on." The shawl is not, however, based on

neg: Golden angle Golden angle In geometry, the golden angle is the smaller of the two angles created by sectioning the circumference of a circle according to the golden ratio; that is, into two arcs such that the ratio of the length of the larger arc to the length of the smaller arc is the same as the ratio of the full circumference to the length of the larger arc. Algebraically, let "a+b" be the circumference of a circle, divided into a longer arc of length "a" and a smaller arc of length "b" such that The golden angle is then the angle

neg: Indiana Pi Bill the chord of 90° as diagonal). Together they describe the circle shown in the figure, whose diameter is 10 and circumference is 32; the chord of 90° is taken to be 7. Both of the values 7 and 32 are within a few percent of the true lengths for a diameter-10 circle (which does not justify Goodwin's presentation of them as exact). The circumference should be nearer to 31.4159 and the diagonal "7" should be the square root of 50 (=25+25), or nearer to 7.071. Goodwin's main goal was not to measure lengths in the circle but to "square" it,

neg: Area of a circle "ds" over the whole circle formula_33 is just the arc length, which is its circumference, so this shows that the area "A" enclosed by the circle is equal to formula_34 times the circumference of the circle. Note that the area of a semicircle of radius r can be computed by the integral formula_35. By trigonometric substitution, we substitute formula_36, hence formula_37 The last step follows since the trigonometric identity formula_42 implies that formula_43 and formula_44 have equal integrals over the interval formula_45, using integration by substitution. But on the other hand, since formula_46, the sum of the two integrals is

neg: Approximations of π circumference of a circle. His 15th-century commentator Nilakantha Somayaji (Kerala school of astronomy and mathematics) has argued that the word means not only that this is an approximation, but that the value is incommensurable (irrational). By the 5th century CE, was known to about seven digits in Chinese mathematics, and to about five in Indian mathematics. Further progress was not made for nearly a millennium, until the 14th century, when Indian mathematician and astronomer Madhava of Sangamagrama, founder of the Kerala school of astronomy and mathematics, discovered the infinite series for , now known as the Madhava–Leibniz series, and gave

neg: Tree volume measurement = volume. Any objection to equation rests primarily with the subjective nature of Fand F. The value 75.4 = 24 π, where 24π substitutes for factor of 12π in the formula for a volume of frustum of a cone encompassing a full tree using one base circumference, converting it to a volume formula that uses a basal circumference that is the average of circumferences C and C By using separate factors for flare and taper and multiplying them together, we create a composite factor. It is suggested that these flare and taper could be extended in some cases to values

neg: Ptolemy's theorem Ptolemy's theorem: The cosine rule for triangle ABC. Let formula_54 Then Therefore, Formula for compound angle sine (+). Let formula_57. Then formula_58. Hence, Formula for compound angle sine (−). This derivation corresponds to the Third Theorem as chronicled by Copernicus following Ptolemy in Almagest. In particular if the sides of a pentagon (subtending 36° at the circumference) and of a hexagon (subtending 30° at the circumference) are given, a chord subtending 6° may be calculated. This was a critical step in the ancient method of calculating tables of chords. This corollary is the core of the Fifth Theorem as chronicled

neg: Kriyakramakari geometric method for computing the value of the circumference of a circle. This technique involves calculating the perimeters of successive regular circumscribed polygons, beginning with a square. Sankara Variar then describes an easier method due to Madhava to compute the value of π. To translate these verses into modern mathematical notations, let C be the circumference and D the diameter of a circle. Then Madhava's easier method to find C reduces to the following expression for C: This is essentially the series known as the Gregory-Leibniz series for π. After stating this series, Sankara Variar follows it up with a

neg: Thales's theorem theorem can also be used to find the centre of a circle using an object with a right angle, such as a set square or rectangular sheet of paper larger than the circle. The angle is placed anywhere on its circumference (figure 1). The intersections of the two sides with the circumference define a diameter (figure 2). Repeating this with a different set of intersections yields another diameter (figure 3). The centre is at the intersection of the diameters. Thales's theorem In geometry, Thales's theorem states that if A, B, and C are distinct points on a circle where the

neg: Transit (astronomy) light curves for stellar host. During a transit there are four "contacts", when the circumference of the small circle (small body disk) touches the circumference of the large circle (large body disk) at a single point. Historically, measuring the precise time of each point of contact was one of the most accurate ways to determine the positions of astronomical bodies. The contacts happen in the following order: A fifth named point is that of greatest transit, when the apparent centers of the two bodies are nearest to each other, halfway through the transit. Since transit photometry allows for scanning large

neg: Joshua Rose (engineer) having a circumference equal to that of the cylinder. The circumference of this circle we divide into any number of equidistant divisions, as from 1 to 24. The pitch A B of the spiral or thread is then divided off also into 24 equidistant divisions, as marked on the left hand of the figure; vertical lines are then drawn from the points of division on the circle to the points correspondingly numbered on the lines dividing the pitch; and where line 1 on the circle intersects line 1 on the pitch is one point in the curve. Similarly, where point

neg: Circumference in terms of these parameters. One such approximation, due to Euler (1773), for the canonical ellipse, is Some lower and upper bounds on the circumference of the canonical ellipse with formula_5 are Here the upper bound formula_9 is the circumference of a circumscribed concentric circle passing through the endpoints of the ellipse's major axis, and the lower bound formula_10 is the perimeter of an inscribed rhombus with vertices at the endpoints of the major and minor axes. The circumference of an ellipse can be expressed exactly in terms of the complete elliptic integral of the second kind. More precisely, we

neg: Hyperbolic tree trees employ hyperbolic space, which intrinsically has "more room" than Euclidean space. For instance, linearly increasing the radius of a circle in Euclidean space increases its circumference linearly, while the same circle in hyperbolic space would have its circumference increase exponentially. Exploiting this property allows laying out the tree in hyperbolic space in an uncluttered manner: placing a node far enough from its parent gives the node almost the same amount of space as its parent for laying out its own children. Displaying a hyperbolic tree commonly utilizes the Poincaré disk model of hyperbolic geometry, though the Klein-Beltrami model can

neg: Jean-Antoine Nollet using insulating silk cords, and electrified, causing his body to accumulate charge. Objects were attracted to him, and close proximity of another person could lead to sparks. In 1746 Nollet gathered about two hundred monks into a circle about a mile (1.6 km) in circumference, with pieces of iron wire connecting them. He then discharged a battery of Leyden jars through the human chain and observed that each man reacted at substantially the same time to the electric shock, showing that the speed of electricity's propagation was very high. In 1748 he discovered the phenomenon of osmosis in natural membranes.

neg: Volte size of the volte. The volte was traditionally 12 strides long (using the inside hind leg) in circumference, resulting in a circle 10–15 meters in diameter. However, this was later decreased to 6-8 strides in circumference. First developed by Saunier, leg-yields are sometimes incorporated into voltes, with the horse's nose facing the inside of the circle. Additionally, de la Guérinière developed "square voltes," which are not commonly practiced today. Riders may incorporate renvers into their volte, as well as use the volte to help prepare the horse for other movements, such as half-pass. Volte The volte is a very small

neg: Earth's circumference shadow on the ground. Using the length of the rod, and the length of the shadow, as the legs of a triangle, he calculated the angle of the sun's rays. This angle was about 7°, or 1/50th the circumference of a circle; taking the Earth as perfectly spherical, he concluded that the Earth's circumference was 50 times the known distance from Alexandra to Syene (5,000 stadia, a figure that was checked yearly), i.e. 250,000 stadia. Depending on whether he used the "Olympic stade" (176.4m) or the Italian stade (184.8m), this would imply a circumference of 44,100 km (an error of

neg: Anthropometry of the upper arm area (UFA), and the arm fat index. The mid-upper arm "area" (MUAA) is an estimation of the area of the upper arm. It is derived from the MUAC using the following formula: The mid-upper arm "muscle" circumference (MUAMC) is an estimation of the circumference of the bone and muscle portions of the upper arm. It is derived from the MUAC and the TSF by accounting for the thickness of the subcutaneous fat that surrounds the muscle, using the following formula, with the MUAC and TSF values measured in millimetres: The mid-upper arm muscle "area" (MUAMA) is an estimation of the

neg: Madhava's sine table (netra : 2), elephants (gaja : 8), snakes (ahi : 8), fires (hutāśana : 3), three (tri : 3), qualities (guṇa : 3), vedas (veda : 4), nakṣatras (bha : 27), elephants (vāraṇa : 8), and arms (bāhavaḥ : 2) - the wise say that this is the measure of the circumference when the diameter of a circle is nava-nikharva (900,000,000,000)." So, the translation of the poem using the bhūtasaṃkhyā system will simply read "2827433388233 is, as the wise say, the circumference of a circle whose diameter is nava-nikharva (900,000,000,000)". That is, divide 2827433388233 (the number from the first two

neg: Unit circle 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

neg: Unit circle 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

neg: Tusi couple which was completed in 1247, that the oscillatory motion be produced by the combined uniform circular motions of two identical circles, one riding on the circumference of the other. At that point, Tusi simply states that if one of these circles were to move at a uniform speed equal to twice the speed of the other, and in a direction to it, then any point on the circumference of the first circle would oscillate in a straight line along one of the diameters of the second circle The term "Tusi couple" is a modern one, coined by Edward Stewart Kennedy

neg: Malebolge guard over them, ready to tear them to pieces. Bolgia Six: Hypocrites are punished in this circle. They are forced to wear heavy lead robes as they walk around the circumference of their circle. The robes are golden and resemble a monk’s cowl but are lined with heavy lead, symbolically representing hypocrisy. Also, Caiaphas, the Pharisee who insisted on the execution of Jesus, is crucified in this circle, staked to the ground so that the ranks of the lead-weighted hypocrites march across him. Bolgia Seven: This bolgia houses the souls of thieves. The bolgia is also filled with serpents, dragons

neg: Sports in Punjab, India circle and the child who has the cloth in his hand goes around the circle-singing: it is a kind of a warning for the children sitting in a circle not to look back. The cloth is then dropped at the back of a child. If it is discovered before the child who had placed it there had completed the round, the child who discovered the cloth would run after him and try to touch him with it till he sits in the place vacated by the one who had discovered the cloth. The song sung during the game is: "Kokla

neg: Earth's circumference Rhodes was 5,000 stadia due north of Alexandria, and the difference in the star's elevation indicated the distance between the two locales was 1/48 of the circle, he multiplied 5,000 by 48 to arrive at a figure of 240,000 stadia for the circumference of the earth. It is generally thought that the stadion used by Posidonius was almost exactly 1/10 of a modern statute mile. Thus Posidonius's measure of 240,000 stadia translates to , not much short of the actual circumference of .<ref name="fragment 202"/ Strabo noted that the distance between Rhodes and Alexandria is 3,750 stadia, and reported Posidonius's

neg: Alabama State Route 210 brother-in-law, who had committed suicide in 1955. Alabama State Route 210 State Route 210 or SR-210, also known as Ross Clark Circle or simply "The Circle" to locals, is a route overlaid by US-84, US-231, and US-431 encircling Dothan, Alabama in Houston County. For many years, the AL-210 designation appeared on state road maps but was not actually signed as such. This has changed in recent years and today the AL-210 designation is noted on signs along with the US Routes that share its circumference of Dothan. SR-210 is highly developed and heavily used both by those who live and

neg: Time Structured Mapping of players who are spatially separated such as with "Four Bridges" which was performed simultaneously in Britain, Germany, America and India, it allows for works which alter the conventional relationship of the composer with the musician by involving the performer directly into the creative process. In 2004 Wyer began creating Simultaneity works: works that made recordings at the same time in different locations. The first recordings took place around Columbus Circle, New York – Wyer mapped out a circumference that passed through the Time Warner building and around the periphery of Columbus Circle itself and, with the help of a

neg: Mesopotamia science was based on a sexagesimal (base 60) numeral system. This is the source of the 60-minute hour, the 24-hour day, and the 360-degree circle. The Sumerian calendar was based on the seven-day week. This form of mathematics was instrumental in early map-making. The Babylonians also had theorems on how to measure the area of several shapes and solids. They measured the circumference of a circle as three times the diameter and the area as one-twelfth the square of the circumference, which would be correct if pi were fixed at 3. The volume of a cylinder was taken as the

neg: Circle a triangle whose base has the length of the circle's circumference and whose height equals the circle's radius, which comes to multiplied by the radius squared: Equivalently, denoting diameter by "d", that is, approximately 79% of the circumscribing square (whose side is of length "d"). The circle is the plane curve enclosing the maximum area for a given arc length. This relates the circle to a problem in the calculus of variations, namely the isoperimetric inequality. Equation of a circle<br> In an "x"–"y" Cartesian coordinate system, the circle with centre coordinates ("a", "b") and radius "r" is the set of

neg: Great circle Great circle A great circle, also known as an orthodrome, of a sphere is the intersection of the sphere and a plane that passes through the center point of the sphere. A great circle is the largest circle that can be drawn on any given sphere. Any diameter of any great circle coincides with a diameter of the sphere, and therefore all great circles have the same center and circumference as each other. This special case of a circle of a sphere is in opposition to a "small circle", that is, the intersection of the sphere and a plane that

neg: Radian Thomson, Muir adopted "radian". As stated, one radian is equal to 180/ degrees. Thus, to convert from radians to degrees, multiply by 180/. For example: Conversely, to convert from degrees to radians, multiply by /180. For example: formula_13 Radians can be converted to turns (complete revolutions) by dividing the number of radians by 2. The length of circumference of a circle is given by formula_14, where formula_15 is the radius of the circle. So the following equivalent relation is true: formula_16[Since a formula_17 sweep is needed to draw a full circle] By the definition of radian, a full circle represents:

neg: Tusi couple Tusi couple The Tusi couple is a mathematical device in which a small circle rotates inside a larger circle twice the diameter of the smaller circle. Rotations of the circles cause a point on the circumference of the smaller circle to oscillate back and forth in linear motion along a diameter of the larger circle. The Tusi couple is a 2-cusped hypocycloid. The couple was first proposed by the 13th-century Persian astronomer and mathematician Nasir al-Din al-Tusi in his 1247 "Tahrir al-Majisti (Commentary on the Almagest)" as a solution for the latitudinal motion of the inferior planets, and later used

neg: Chromatic circle direction moved. Continuing from that point (across the circle and one tick clockwise) all points are connected. Moving through this pattern the notes of the circle of fifths can be determined (C, G, D, A ...). The ratio of the frequencies between two pitches in the constellation can be determined as follows. Take the length of the arc (measured clockwise) between the two points and divide by the circumference of the circle. The frequency ratio is two raised to this power. For example, for a fifth (P5, which is located at 7 o'clock relative to the tonic T) the frequency

neg: Aristotle's wheel paradox rolls) on the curb while your tire merely rolls on the pavement. The smaller the small circle relative to the large circle, the more the small one slips. Of course the center of the two circles does not rotate at all, so it slides the whole way. Alternatively, one can reject the assumption that the smaller circle is independent of the larger circle. Imagine a tire as the larger circle, and imagine the smaller circle as the interior circumference of the tire and not as the rim. The movement of the inner circle is dependent on the larger circle. Thus

neg: Isosceles triangle circle can be partitioned into isosceles triangles by the radii of this circle through its vertices. The fact that all radii of a circle have equal length implies that all of these triangles are isosceles. This partition can be used to derive a formula for the area of the polygon as a function of its side lengths, even for cyclic polygons that do not contain their circumcenters. This formula generalizes Heron's formula for triangles and Brahmagupta's formula for cyclic quadrilaterals. Either diagonal of a rhombus divides it into two congruent isosceles triangles. Similarly, one of the two diagonals of a

neg: Alabama State Route 210 Alabama State Route 210 State Route 210 or SR-210, also known as Ross Clark Circle or simply "The Circle" to locals, is a route overlaid by US-84, US-231, and US-431 encircling Dothan, Alabama in Houston County. For many years, the AL-210 designation appeared on state road maps but was not actually signed as such. This has changed in recent years and today the AL-210 designation is noted on signs along with the US Routes that share its circumference of Dothan. SR-210 is highly developed and heavily used both by those who live and work in Dothan and by motorists bound

neg: Ellipse "Modular Equations and Approximations to formula_265"; they are and The errors in these approximations, which were obtained empirically, are of order formula_378 and formula_379 respectively. More generally, the arc length of a portion of the circumference, as a function of the angle subtended, is given by an incomplete elliptic integral. The inverse function, the angle subtended as a function of the arc length, is given by the elliptic functions. Some lower and upper bounds on the circumference of the canonical ellipse formula_380 with formula_381 are Here the upper bound formula_385 is the circumference of a circumscribed concentric circle passing through

neg: Area the circle. This argument is actually a simple application of the ideas of calculus. In ancient times, the method of exhaustion was used in a similar way to find the area of the circle, and this method is now recognized as a precursor to integral calculus. Using modern methods, the area of a circle can be computed using a definite integral: The formula for the area enclosed by an ellipse is related to the formula of a circle; for an ellipse with semi-major and semi-minor axes and the formula is: Most basic formulas for surface area can be obtained by