Volume is the quantity of three-dimensional space enclosed by a closed surface, for example, the space that a substance or shape occupies or contains. Volume is quantified numerically using the SI derived unit, the cubic metre; the volume of a container is understood to be the capacity of the container. Three dimensional mathematical shapes are assigned volumes. Volumes of some simple shapes, such as regular, straight-edged, circular shapes can be calculated using arithmetic formulas. Volumes of complicated shapes can be calculated with integral calculus if a formula exists for the shape's boundary. One-dimensional figures and two-dimensional shapes are assigned zero volume in the three-dimensional space; the volume of a solid can be determined by fluid displacement. Displacement of liquid can be used to determine the volume of a gas; the combined volume of two substances is greater than the volume of just one of the substances. However, sometimes one substance dissolves in the other and in such cases the combined volume is not additive.
In differential geometry, volume is expressed by means of the volume form, is an important global Riemannian invariant. In thermodynamics, volume is a fundamental parameter, is a conjugate variable to pressure. Any unit of length gives a corresponding unit of volume: the volume of a cube whose sides have the given length. For example, a cubic centimetre is the volume of a cube. In the International System of Units, the standard unit of volume is the cubic metre; the metric system includes the litre as a unit of volume, where one litre is the volume of a 10-centimetre cube. Thus 1 litre = 3 = 1000 cubic centimetres = 0.001 cubic metres. Small amounts of liquid are measured in millilitres, where 1 millilitre = 0.001 litres = 1 cubic centimetre. In the same way, large amounts can be measured in megalitres, where 1 million litres = 1000 cubic metres = 1 megalitre. Various other traditional units of volume are in use, including the cubic inch, the cubic foot, the cubic yard, the cubic mile, the teaspoon, the tablespoon, the fluid ounce, the fluid dram, the gill, the pint, the quart, the gallon, the minim, the barrel, the cord, the peck, the bushel, the hogshead, the acre-foot and the board foot.
Capacity is defined by the Oxford English Dictionary as "the measure applied to the content of a vessel, to liquids, grain, or the like, which take the shape of that which holds them". Capacity is not identical in meaning to volume, though related. Units of capacity are the SI litre and its derived units, Imperial units such as gill, pint and others. Units of volume are the cubes of units of length. In SI the units of volume and capacity are related: one litre is 1 cubic decimetre, the capacity of a cube with a 10 cm side. In other systems the conversion is not trivial; the density of an object is defined as the ratio of the mass to the volume. The inverse of density is specific volume, defined as volume divided by mass. Specific volume is a concept important in thermodynamics where the volume of a working fluid is an important parameter of a system being studied; the volumetric flow rate in fluid dynamics is the volume of fluid which passes through a given surface per unit time. In calculus, a branch of mathematics, the volume of a region D in R3 is given by a triple integral of the constant function f = 1 and is written as: ∭ D 1 d x d y d z.
The volume integral in cylindrical coordinates is ∭ D r d r d θ d z, the volume integral in spherical coordinates has the form ∭ D ρ 2 sin ϕ d ρ d θ d ϕ. The above formulas can be used to show that the volumes of a cone and cylinder of the same radius and height are in the ratio 1: 2: 3, as follows. Let the radius be r and the height be h the volume of cone is 1 3 π r 2 h = 1 3 π r 2 = × 1, the volume of the sphere
In mathematics, a negative number is a real number, less than zero. Negative numbers represent opposites. If positive represents a movement to the right, negative represents a movement to the left. If positive represents above sea level negative represents below sea level. If positive represents a deposit, negative represents a withdrawal, they are used to represent the magnitude of a loss or deficiency. A debt, owed may be thought of as a negative asset, a decrease in some quantity may be thought of as a negative increase. If a quantity may have either of two opposite senses one may choose to distinguish between those senses—perhaps arbitrarily—as positive and negative. In the medical context of fighting a tumor, an expansion could be thought of as a negative shrinkage. Negative numbers are used to describe values on a scale that goes below zero, such as the Celsius and Fahrenheit scales for temperature; the laws of arithmetic for negative numbers ensure that the common sense idea of an opposite is reflected in arithmetic.
For example, − = 3 because the opposite of an opposite is the original value. Negative numbers are written with a minus sign in front. For example, −3 represents a negative quantity with a magnitude of three, is pronounced "minus three" or "negative three". To help tell the difference between a subtraction operation and a negative number the negative sign is placed higher than the minus sign. Conversely, a number, greater than zero is called positive; the positivity of a number may be emphasized by placing a plus sign before it, e.g. +3. In general, the negativity or positivity of a number is referred to as its sign; every real number other than zero is either negative. The positive whole numbers are referred to as natural numbers, while the positive and negative whole numbers are referred to as integers. In bookkeeping, amounts owed are represented by red numbers, or a number in parentheses, as an alternative notation to represent negative numbers. Negative numbers appeared for the first time in history in the Nine Chapters on the Mathematical Art, which in its present form dates from the period of the Chinese Han Dynasty, but may well contain much older material.
Liu Hui established rules for subtracting negative numbers. By the 7th century, Indian mathematicians such as Brahmagupta were describing the use of negative numbers. Islamic mathematicians further developed the rules of subtracting and multiplying negative numbers and solved problems with negative coefficients. Western mathematicians accepted the idea of negative numbers around the middle of the 19th century. Prior to the concept of negative numbers, mathematicians such as Diophantus considered negative solutions to problems "false" and equations requiring negative solutions were described as absurd; some mathematicians like Leibniz agreed that negative numbers were false, but still used them in calculations. Negative numbers can be thought of as resulting from the subtraction of a larger number from a smaller. For example, negative three is the result of subtracting three from zero: 0 − 3 = −3. In general, the subtraction of a larger number from a smaller yields a negative result, with the magnitude of the result being the difference between the two numbers.
For example, 5 − 8 = −3since 8 − 5 = 3. The relationship between negative numbers, positive numbers, zero is expressed in the form of a number line: Numbers appearing farther to the right on this line are greater, while numbers appearing farther to the left are less, thus zero appears in the middle, with the positive numbers to the right and the negative numbers to the left. Note that a negative number with greater magnitude is considered less. For example though 8 is greater than 5, written 8 > 5negative 8 is considered to be less than negative 5: −8 < −5. It follows that any negative number is less than any positive number, so −8 < 5 and −5 < 8. In the context of negative numbers, a number, greater than zero is referred to as positive, thus every real number other than zero is either positive or negative, while zero itself is not considered to have a sign. Positive numbers are sometimes written with a plus sign in front, e.g. +3 denotes a positive three. Because zero is neither positive nor negative, the term nonnegative is sometimes used to refer to a number, either positive or zero, while nonpositive is used to refer to a number, either negative or zero.
Zero is a neutral number. Goal difference in association football and hockey. Plus-minus differential in ice hockey: the difference in total goals scored for the team and against the team when a particular player is on the ice is the player’s +/− rating. Players can have a negative rating. Run differential in baseball: the run differential is negative if the team allows more runs than they scored. British football clubs are deducted points if they enter administration, thus have a negative points total until they have earned at least that many points that season. Lap times in Formula 1 may be given as the difference compared to a previous lap, will be positive if slower and negative if faster. In some athletics events, such as sprint races, the hurdles, the triple jump and the long jump, the wind assistance is measured and recorde
In geometry, the hyperboloid model known as the Minkowski model or the Lorentz model, is a model of n-dimensional hyperbolic geometry in which points are represented by the points on the forward sheet S+ of a two-sheeted hyperboloid in -dimensional Minkowski space and m-planes are represented by the intersections of the -planes in Minkowski space with S+. The hyperbolic distance function admits a simple expression in this model; the hyperboloid model of the n-dimensional hyperbolic space is related to the Beltrami–Klein model and to the Poincaré disk model as they are projective models in the sense that the isometry group is a subgroup of the projective group. If is a vector in the -dimensional coordinate space Rn+1, the Minkowski quadratic form is defined to be Q = x 0 2 − x 1 2 − … − x n 2; the vectors v ∈ Rn+1 such that Q = 1 form an n-dimensional hyperboloid S consisting of two connected components, or sheets: the forward, or future, sheet S+, where x0>0 and the backward, or past, sheet S−, where x0<0.
The points of the n-dimensional hyperboloid model are the points on the forward sheet S+. The Minkowski bilinear form B is the polarization of the Minkowski quadratic form Q, B = / 2. Explicitly, B = x 0 y 0 − x 1 y 1 − … − x n y n; the hyperbolic distance between two points u and v of S+ is given by the formula d = arcosh , where arcosh is the inverse function of hyperbolic cosine. A straight line in hyperbolic n-space is modeled by a geodesic on the hyperboloid. A geodesic on the hyperboloid is the intersection of the hyperboloid with a two-dimensional linear subspace of the n+1-dimensional Minkowski space. If we take u and v to be basis vectors of that linear subspace with B = 1 B = − 1 B = B = 0 and use w as a real parameter for points on the geodesic u cosh w + v sinh w will be a point on the geodesic. More a k-dimensional "flat" in the hyperbolic n-space will be modeled by the intersection of the hyperboloid with a k+1-dimensional linear subspace of the Minkowski space; the indefinite orthogonal group O called the -dimensional Lorentz group, is the Lie group of real × matrices which preserve the Minkowski bilinear form.
In a different language, it is the group of linear isometries of the Minkowski space. In particular, this group preserves the hyperboloid S. Recall that indefinite orthogonal groups have four connected components, corresponding to reversing or preserving the orientation on each subspace, form a Klein four-group; the subgroup of O that preserves the sign of the first coordinate is the orthochronous Lorentz group, denoted O+, has two components, corresponding to preserving or reversing the orientation of the spatial subspace. Its subgroup SO+ consisting of matrices with determinant one is a connected Lie group of dimension n/2 which acts on S+ by linear automorphisms and preserves the hyperbolic distance; this action is transitive and the stabilizer of the vector consists of the matrices of the form Where A belongs to the compact special orthogonal group SO (generalizing the rotation group SO
Eugenio Beltrami was an Italian mathematician notable for his work concerning differential geometry and mathematical physics. His work was noted for clarity of exposition, he was the first to prove consistency of non-Euclidean geometry by modeling it on a surface of constant curvature, the pseudosphere, in the interior of an n-dimensional unit sphere, the so-called Beltrami–Klein model. He developed singular value decomposition for matrices, subsequently rediscovered several times. Beltrami's use of differential calculus for problems of mathematical physics indirectly influenced development of tensor calculus by Gregorio Ricci-Curbastro and Tullio Levi-Civita. Beltrami was born in Cremona in Lombardy a part of the Austrian Empire, now part of Italy, he began studying mathematics at University of Pavia in 1853, but was expelled from Ghislieri College in 1856 due to his political opinions—he was sympathetic with the Risorgimento. During this time he was influenced by Francesco Brioschi, he had to discontinue his studies because of financial hardship and spent the next several years as a secretary working for the Lombardy–Venice railroad company.
He was appointed to the University of Bologna as a professor in 1862, the year he published his first research paper. Throughout his life, Beltrami had various professorial jobs at the universities of Pisa and Pavia. From 1891 until the end of his life Beltrami lived in Rome, he became the president of the Accademia dei Lincei in 1898 and a senator of the Kingdom of Italy in 1899. In 1868 Beltrami published two memoirs dealing with consistency and interpretations of non-Euclidean geometry of János Bolyai and Nikolai Lobachevsky. In his "Essay on an interpretation of non-Euclidean geometry", Beltrami proposed that this geometry could be realized on a surface of constant negative curvature, a pseudosphere. For Beltrami's concept, lines of the geometry are represented by geodesics on the pseudosphere and theorems of non-Euclidean geometry can be proved within ordinary three-dimensional Euclidean space, not derived in an axiomatic fashion, as Lobachevsky and Bolyai had done previously. In 1840, Ferdinand Minding considered geodesic triangles on the pseudosphere and remarked that the corresponding "trigonometric formulas" are obtained from the corresponding formulas of spherical trigonometry by replacing the usual trigonometric functions with hyperbolic functions.
In this way, Beltrami attempted to demonstrate that two-dimensional non-Euclidean geometry is as valid as the Euclidean geometry of the space, in particular, that Euclid's parallel postulate could not be derived from the other axioms of Euclidean geometry. It is stated that this proof was incomplete due to the singularities of the pseudosphere, which means that geodesics could not be extended indefinitely. However, John Stillwell remarks that Beltrami must have been well aware of this difficulty, manifested by the fact that the pseudosphere is topologically a cylinder, not a plane, he spent a part of his memoir designing a way around it. By a suitable choice of coordinates, Beltrami showed how the metric on the pseudosphere can be transferred to the unit disk and that the singularity of the pseudosphere corresponds to a horocycle on the non-Euclidean plane. On the other hand, in the introduction to his memoir, Beltrami states that it would be impossible to justify "the rest of Lobachevsky's theory", i.e. the non-Euclidean geometry of space, by this method.
In the second memoir published during the same year, "Fundamental theory of spaces of constant curvature", Beltrami continued this logic and gave an abstract proof of equiconsistency of hyperbolic and Euclidean geometry for any dimension. He accomplished this by introducing several models of non-Euclidean geometry that are now known as the Beltrami–Klein model, the Poincaré disk model, the Poincaré half-plane model, together with transformations that relate them. For the half-plane model, Beltrami cited a note by Joseph Liouville in the treatise of Gaspard Monge on differential geometry. Beltrami showed that n-dimensional Euclidean geometry is realized on a horosphere of the -dimensional hyperbolic space, so the logical relation between consistency of the Euclidean and the non-Euclidean geometries is symmetric. Beltrami acknowledged the influence of Bernhard Riemann's groundbreaking Habilitation lecture "On the hypotheses on which geometry is based". Although today Beltrami's "Essay" is recognized as important for the development of non-Euclidean geometry, the reception at the time was less enthusiastic.
Luigi Cremona objected to perceived circular reasoning, which forced Beltrami to delay the publication of the "Essay" by one year. Subsequently, Felix Klein failed to acknowledge Beltrami's priority in construction of the projective disk model of the non-Euclidean geometry; this reaction can be attributed in part to the novelty of Beltrami's reasoning, similar to the ideas of Riemann concerning abstract manifolds. J. Hoüel published Beltrami's proof in his French translation of works of Bolyai. Beltrami, Eugenio. "Saggio di interpretazione della geometria non-euclidea". Giornale di Mathematiche. VI: 285–315. Beltrami, Eugenio. "Teoria fondamentale degli spazii di curvatura costante". Annali. Di Mat. ser II. 2: 232–255. Doi:10.1007/BF02419615. Opere matematiche di Eugenio Beltrami pubblicate per cura della Facoltà di scienze della r. Università di Roma (U. Hoepli, Mila
University of Manchester
The University of Manchester is a public research university in Manchester, formed in 2004 by the merger of the University of Manchester Institute of Science and Technology and the Victoria University of Manchester. The University of Manchester is a red brick university, a product of the civic university movement of the late 19th century; the main campus is south of Manchester city centre on Oxford Road. In 2016/17, the university had 40,490 students and 10,400 staff, making it the second largest university in the UK, the largest single-site university; the university had a consolidated income of £1 billion in 2017–18, of which £298.7 million was from research grants and contracts. It has the fourth-largest endowment of any university in the UK, after the universities of Cambridge and Edinburgh, it is a member of the worldwide Universities Research Association, the Russell Group of British research universities and the N8 Group. For 2018–19, the University of Manchester was ranked 29th in the world and 6th in the UK by QS World University Rankings.
In 2017 it was ranked 38th in the world and 6th in the UK by Academic Ranking of World Universities, 55th in the world and 8th in the UK by Times Higher Education World University Rankings and 59th in the world by U. S. News and World Report. Manchester was ranked 15th in the UK amongst multi-faculty institutions for the quality of its research and 5th for its Research Power in the 2014 Research Excellence Framework; the university owns and operates major cultural assets such as the Manchester Museum, Whitworth Art Gallery, John Rylands Library and Jodrell Bank Observatory and its Grade I listed Lovell Telescope. The University of Manchester has 25 Nobel laureates among its past and present students and staff, the fourth-highest number of any single university in the United Kingdom. Four Nobel laureates are among its staff – more than any other British university; the University of Manchester traces its roots to the formation of the Mechanics' Institute in 1824, its heritage is linked to Manchester's pride in being the world's first industrial city.
The English chemist John Dalton, together with Manchester businessmen and industrialists, established the Mechanics' Institute to ensure that workers could learn the basic principles of science. John Owens, a textile merchant, left a bequest of £96,942 in 1846 to found a college to educate men on non-sectarian lines, his trustees established Owens College in 1851 in a house on the corner of Quay Street and Byrom Street, the home of the philanthropist Richard Cobden, subsequently housed Manchester County Court. The locomotive designer, Charles Beyer became a governor of the college and was the largest single donor to the college extension fund, which raised the money to move to a new site and construct the main building now known as the John Owens building, he campaigned and helped fund the engineering chair, the first applied science department in the north of England. He left the college the equivalent of £10 million in his will in 1876, at a time when it was in great financial difficulty.
Beyer funded the total cost of construction of the Beyer building to house the biology and geology departments. His will funded Engineering chairs and the Beyer Professor of Applied mathematics; the university has a rich German heritage. The Owens College Extension Movement based their plans after a tour of German universities and polytechnics. Manchester mill owner, Thomas Ashton, chairman of the extension movement had studied at Heidelberg University. Sir Henry Roscoe studied at Heidelberg under Robert Bunsen and they collaborated for many years on research projects. Roscoe promoted the German style of research led teaching that became the role model for the redbrick universities. Charles Beyer studied at Dresden Academy Polytechnic. There were many Germans on the staff, including Carl Schorlemmer, Britain's first chair in organic chemistry, Arthur Schuster, professor of Physics. There was a German chapel on the campus. In 1873 the college moved to new premises on Oxford Road, Chorlton-on-Medlock and from 1880 it was a constituent college of the federal Victoria University.
The university was established and granted a Royal Charter in 1880 becoming England's first civic university. By 1905, the institutions were active forces; the Municipal College of Technology, forerunner of UMIST, was the Victoria University of Manchester's Faculty of Technology while continuing in parallel as a technical college offering advanced courses of study. Although UMIST achieved independent university status in 1955, the universities continued to work together. However, in the late-20th century, formal connections between the university and UMIST diminished and in 1994 most of the remaining institutional ties were severed as new legislation allowed UMIST to become an autonomous university with powers to award its own degrees. A decade the development was reversed; the Victoria University of Manchester and the University of Manchester Institute of Science and Technology agreed to merge into a single institution in March 2003. Before the merger, Victoria University of Manchester and UMIST counted 23 Nobel Prize winners amongst their former staff and students, with two further Nobel laureates being subsequently added.
Manchester has traditionally been strong in the sciences. Notable scientists as
Gabriel's horn is a geometric figure which has infinite surface area but finite volume. The name refers to the Abrahamic tradition identifying the archangel Gabriel as the angel who blows the horn to announce Judgment Day, associating the divine, or infinite, with the finite; the properties of this figure were first studied by Italian physicist and mathematician Evangelista Torricelli in the 17th century. Gabriel's horn is formed by taking the graph of x ↦ 1 x, with the domain x ≥ 1 and rotating it in three dimensions about the x-axis; the discovery was made using Cavalieri's principle before the invention of calculus, but today calculus can be used to calculate the volume and surface area of the horn between x = 1 and x = a, where a > 1. Using integration, it is possible to find the volume V and the surface area A: V = π ∫ 1 a 2 d x = π A = 2 π ∫ 1 a 1 x 1 + 2 d x > 2 π ∫ 1 a d x x = 2 π ln . The value a can be as large as required, but it can be seen from the equation that the volume of the part of the horn between x = 1 and x = a will never exceed π.
Mathematically, the volume approaches π as a approaches infinity. Using the limit notation of calculus: lim a → ∞ V = lim a → ∞ π = π ⋅ lim a → ∞ = π; the surface area formula above gives a lower bound for the area as 2π times the natural logarithm of a. There is no upper bound for the natural logarithm of a, as a approaches infinity; that means, in this case. That is to say, lim a → ∞ A ≥ lim a → ∞ 2 π ln = ∞; when the properties of Gabriel's horn were discovered, the fact that the rotation of an infinitely large section of the xy-plane about the x-axis generates an object of finite volume was considered paradoxical. While the section lying in the xy-plane has an infinite area, any other section parallel to it has a finite area, thus the volume, being calculated from the "weighted sum" of sections, is finite. Another approach is to treat the horn as a stack of disks with diminishing radii; the sum of the radii produces a harmonic series. However, the correct calculation is the sum of their squares.
Every disk has a radius r = 1/x and an area πr2 or π/x2. The series 1/x diverges but 1/x2 converges. In general, for any real ε > 0, 1/x1+ε converges. The apparent paradox formed part of a dispute over the nature of infinity involving many of the key thinkers of the time including Thomas Hobbes, John Wallis and Galileo Galilei. There is a similar phenomenon which applies to areas in the plane; the area between the curves 1/x2 and -1/x2 from 1 to infinity is finite, but the lengths of the two curves are infinite. Since the horn has finite volume but infinite surface area, there is an apparent paradox that the horn could be filled with a finite quantity of paint and yet that paint would not be sufficient to coat its inner surface; the paradox is resolved by realizing that a finite amount of paint can in fact coat an infinite surface area — it needs to get thinner at a fast enough rate. In the case where the horn is filled with paint, this thinning is accomplished by the increasing reduction in diameter of the throat of the horn.
The converse of Gabriel's horn—a surface of revolution that has a finite surface area but an infinite volume—cannot occur: Let f: [1,∞) → [0,∞) be a continuously differentiable function. Write S for the solid of revolution of the graph y = f about the x-axis. If the surface area of S is finite so is the volume. Since the lateral surface area A is finite, the limit superior: lim t → ∞ sup x
Area is the quantity that expresses the extent of a two-dimensional figure or shape, or planar lamina, in the plane. Surface area is its analog on the two-dimensional surface of a three-dimensional object. Area can be understood as the amount of material with a given thickness that would be necessary to fashion a model of the shape, or the amount of paint necessary to cover the surface with a single coat, it is the two-dimensional analog of the volume of a solid. The area of a shape can be measured by comparing the shape to squares of a fixed size. In the International System of Units, the standard unit of area is the square metre, the area of a square whose sides are one metre long. A shape with an area of three square metres would have the same area as three such squares. In mathematics, the unit square is defined to have area one, the area of any other shape or surface is a dimensionless real number. There are several well-known formulas for the areas of simple shapes such as triangles and circles.
Using these formulas, the area of any polygon can be found by dividing the polygon into triangles. For shapes with curved boundary, calculus is required to compute the area. Indeed, the problem of determining the area of plane figures was a major motivation for the historical development of calculus. For a solid shape such as a sphere, cone, or cylinder, the area of its boundary surface is called the surface area. Formulas for the surface areas of simple shapes were computed by the ancient Greeks, but computing the surface area of a more complicated shape requires multivariable calculus. Area plays an important role in modern mathematics. In addition to its obvious importance in geometry and calculus, area is related to the definition of determinants in linear algebra, is a basic property of surfaces in differential geometry. In analysis, the area of a subset of the plane is defined using Lebesgue measure, though not every subset is measurable. In general, area in higher mathematics is seen as a special case of volume for two-dimensional regions.
Area can be defined through the use of axioms, defining it as a function of a collection of certain plane figures to the set of real numbers. It can be proved. An approach to defining what is meant by "area" is through axioms. "Area" can be defined as a function from a collection M of special kind of plane figures to the set of real numbers which satisfies the following properties: For all S in M, a ≥ 0. If S and T are in M so are S ∪ T and S ∩ T, a = a + a − a. If S and T are in M with S ⊆ T T − S is in M and a = a − a. If a set S is in M and S is congruent to T T is in M and a = a; every rectangle R is in M. If the rectangle has length h and breadth k a = hk. Let Q be a set enclosed between two step regions S and T. A step region is formed from a finite union of adjacent rectangles resting on a common base, i.e. S ⊆ Q ⊆ T. If there is a unique number c such that a ≤ c ≤ a for all such step regions S and T a = c, it can be proved that such an area function exists. Every unit of length has a corresponding unit of area, namely the area of a square with the given side length.
Thus areas can be measured in square metres, square centimetres, square millimetres, square kilometres, square feet, square yards, square miles, so forth. Algebraically, these units can be thought of as the squares of the corresponding length units; the SI unit of area is the square metre, considered an SI derived unit. Calculation of the area of a square whose length and width are 1 metre would be: 1 metre x 1 metre = 1 m2and so, a rectangle with different sides would have an area in square units that can be calculated as: 3 metres x 2 metres = 6 m2; this is equivalent to 6 million square millimetres. Other useful conversions are: 1 square kilometre = 1,000,000 square metres 1 square metre = 10,000 square centimetres = 1,000,000 square millimetres 1 square centimetre = 100 square millimetres. In non-metric units, the conversion between two square units is the square of the conversion between the corresponding length units. 1 foot = 12 inches,the relationship between square feet and square inches is 1 square foot = 144 square inches,where 144 = 122 = 12 × 12.
Similarly: 1 square yard = 9 square feet 1 square mile = 3,097,600 square yards = 27,878,400 square feetIn addition, conversion factors include: 1 square inch = 6.4516 square centimetres 1 square foot = 0.09290304 square metres 1 square yard = 0.83612736 square metres 1 square mile = 2.589988110336 square kilometres There are several other common units for area. The are was the original unit of area in the metric system, with: 1 are = 100 square metresThough the are has fallen out of use, the hectare is still used to measure land: 1 hectare = 100 ares = 10,000 square metres = 0.01 square kilometresOther uncommon metric units of area include the tetrad, the hectad, the myriad. The acre is commonly used to measure land areas, where 1 acre = 4,840 square yards = 43,560 square feet. An acre is 40% of a hectare. On the atomic scale, area is measured in units of barns, such that: 1 barn = 10−28 square meters; the barn is used in describing the cross-sectional area of interaction in nuclear physics.
In India, 20 dhurki = 1 dhur 20 dhur = 1 khatha 20 khata = 1 bigha 32 khata = 1 acre In the 5th century BCE, Hippocrates of Chios was the first to show that the area of a disk is proportional to the square of its diameter, as part of his quadrature of the lune of