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50-622: The district has a total land area of 12.54 km. Its administrative center is located 197 meters above sea level . Initially, the boundary with Ate was delineated by the Río Surco irrigation ditch. However, in 1989, the eastern section of El Agustino, situated east of the El Agustino hill, separated to establish the Santa Anita district. This newly formed district incorporated the Santa Anita section of Ate along with surrounding areas. According to

100-567: A , b ) {\displaystyle I=(a,b)} , in the set R {\displaystyle \mathbb {R} } of real numbers, let ℓ ( I ) = b − a {\displaystyle \ell (I)=b-a} denote its length. For any subset E ⊆ R {\displaystyle E\subseteq \mathbb {R} } , the Lebesgue outer measure λ ∗ ( E ) {\displaystyle \lambda ^{\!*\!}(E)}

150-664: A σ -algebra . A set E {\displaystyle E} that does not satisfy the Carathéodory criterion is not Lebesgue-measurable. ZFC proves that non-measurable sets do exist; an example is the Vitali sets . The first part of the definition states that the subset E {\displaystyle E} of the real numbers is reduced to its outer measure by coverage by sets of open intervals. Each of these sets of intervals I {\displaystyle I} covers E {\displaystyle E} in

200-400: 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 x and y the formula is: Most basic formulas for surface area can be obtained by cutting surfaces and flattening them out (see: developable surfaces ). For example, if the side surface of a cylinder (or any prism )

250-423: A "nicer" set B which differs from A only by a null set (in the sense that the symmetric difference ( A − B ) ∪ ( B − A ) is a null set) and then show that B can be generated using countable unions and intersections from open or closed sets. The modern construction of the Lebesgue measure is an application of Carathéodory's extension theorem . It proceeds as follows. Fix n ∈ N . A box in R

300-508: A 2002 estimate by the INEI , the district has 166,177 inhabitants and a population density of 13,251.8 persons/km. In 1999, there were 32,910 households in the district. It is the 25th most populated district in Lima. This Lima Region geography article is a stub . You can help Misplaced Pages by expanding it . Area Area is the measure of a region 's size on a surface . The area of

350-517: A Lebesgue measure are called Lebesgue-measurable ; the measure of the Lebesgue-measurable set A is here denoted by λ ( A ). Henri Lebesgue described this measure in the year 1901 which, a year after, was followed up by his description of the Lebesgue integral. Both were published as part of his dissertation in 1902. For any interval I = [ a , b ] {\displaystyle I=[a,b]} , or I = (

400-417: A corresponding unit of area, namely the area of a square with the given side length. Thus areas can be measured in square metres (m ), square centimetres (cm ), square millimetres (mm ), square kilometres (km ), square feet (ft ), square yards (yd ), square miles (mi ), and so forth. Algebraically, these units can be thought of as the squares of the corresponding length units. The SI unit of area

450-405: A plane region or plane area refers to the area of a shape or planar lamina , while surface area refers to the area of an open surface or the boundary 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

500-428: A rectangle with length l and width w , the formula for the area is: That is, the area of the rectangle is the length multiplied by the width. As a special case, as l = w in the case of a square, the area of a square with side length s is given by the formula: The formula for the area of a rectangle follows directly from the basic properties of area, and is sometimes taken as a definition or axiom . On

550-468: A sense, since the union of these intervals contains E {\displaystyle E} . The total length of any covering interval set may overestimate the measure of E , {\displaystyle E,} because E {\displaystyle E} is a subset of the union of the intervals, and so the intervals may include points which are not in E {\displaystyle E} . The Lebesgue outer measure emerges as

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600-433: 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 usually requires multivariable calculus . Area plays an important role in modern mathematics. In addition to its obvious importance in geometry and calculus, area

650-525: A sphere was first obtained by Archimedes in his work On the Sphere and Cylinder . The formula is: where r is the radius of the sphere. As with the formula for the area of a circle, any derivation of this formula inherently uses methods similar to calculus . Lebesgue measure In measure theory , a branch of mathematics , the Lebesgue measure , named after French mathematician Henri Lebesgue ,

700-405: Is a set of the form where b i ≥ a i , and the product symbol here represents a Cartesian product. The volume of this box is defined to be For any subset A of R , we can define its outer measure λ *( A ) by: We then define the set A to be Lebesgue-measurable if for every subset S of R , These Lebesgue-measurable sets form a σ -algebra , and the Lebesgue measure

750-447: Is approximately triangular in shape, and the sectors can be rearranged to form an approximate parallelogram. The height of this parallelogram is r , and the width is half the circumference of the circle, or π r . Thus, the total area of the circle is π r : 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

800-418: Is cut lengthwise, the surface can be flattened out into a rectangle. Similarly, if a cut is made along the side of a cone , the side surface can be flattened out into a sector of a circle, and the resulting area computed. The formula for the surface area of a sphere is more difficult to derive: because a sphere has nonzero Gaussian curvature , it cannot be flattened out. The formula for the surface area of

850-891: Is defined as an infimum The above definition can be generalised to higher dimensions as follows. For any rectangular cuboid C {\displaystyle C} which is a Cartesian product C = I 1 × ⋯ × I n {\displaystyle C=I_{1}\times \cdots \times I_{n}} of open intervals, let vol ⁡ ( C ) = ℓ ( I 1 ) × ⋯ × ℓ ( I n ) {\displaystyle \operatorname {vol} (C)=\ell (I_{1})\times \cdots \times \ell (I_{n})} (a real number product) denote its volume. For any subset E ⊆ R n {\displaystyle E\subseteq \mathbb {R^{n}} } , Some sets E {\displaystyle E} satisfy

900-410: Is defined by λ ( A ) = λ *( A ) for any Lebesgue-measurable set A . The existence of sets that are not Lebesgue-measurable is a consequence of the set-theoretical axiom of choice , which is independent from many of the conventional systems of axioms for set theory . The Vitali theorem , which follows from the axiom, states that there exist subsets of R that are not Lebesgue-measurable. Assuming

950-462: Is known as Heron's formula for the area of a triangle in terms of its sides, and a proof can be found in his book, Metrica , written around 60 CE. It has been suggested that Archimedes knew the formula over two centuries earlier, and since Metrica is a collection of the mathematical knowledge available in the ancient world, it is possible that the formula predates the reference given in that work. In 300 BCE Greek mathematician Euclid proved that

1000-529: Is not in E {\displaystyle E} : the set difference of A {\displaystyle A} and E {\displaystyle E} . These partitions of A {\displaystyle A} are subject to the outer measure. If for all possible such subsets A {\displaystyle A} of the real numbers, the partitions of A {\displaystyle A} cut apart by E {\displaystyle E} have outer measures whose sum

1050-431: Is related to the definition of determinants in linear algebra , and 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 if one supposes the axiom of choice. In general, area in higher mathematics is seen as a special case of volume for two-dimensional regions. Area can be defined through

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1100-442: Is tested by taking subsets A {\displaystyle A} of the real numbers using E {\displaystyle E} as an instrument to split A {\displaystyle A} into two partitions: the part of A {\displaystyle A} which intersects with E {\displaystyle E} and the remaining part of A {\displaystyle A} which

1150-410: Is the outer measure of A {\displaystyle A} , then the outer Lebesgue measure of E {\displaystyle E} gives its Lebesgue measure. Intuitively, this condition means that the set E {\displaystyle E} must not have some curious properties which causes a discrepancy in the measure of another set when E {\displaystyle E}

1200-470: Is the square metre, which is considered an SI derived unit . Calculation of the area of a square whose length and width are 1 metre would be: 1 metre × 1 metre = 1 m and so, a rectangle with different sides (say length of 3 metres and width of 2 metres) would have an area in square units that can be calculated as: 3 metres × 2 metres = 6 m . This is equivalent to 6 million square millimetres. Other useful conversions are: In non-metric units,

1250-436: Is the standard way of assigning a measure to subsets of higher dimensional Euclidean n -spaces . For lower dimensions n = 1, 2, or 3, it coincides with the standard measure of length , area , or volume . In general, it is also called n -dimensional volume , n -volume , hypervolume , or simply volume . It is used throughout real analysis , in particular to define Lebesgue integration . Sets that can be assigned

1300-411: Is the two-dimensional analogue of the length of a curve (a one-dimensional concept) or the volume of a solid (a three-dimensional concept). Two different regions may have the same area (as in squaring the circle ); by synecdoche , "area" sometimes is used to refer to the region, as in a " polygonal area ". The area of a shape can be measured by comparing the shape to squares of a fixed size. In

1350-423: Is used as a "mask" to "clip" that set, hinting at the existence of sets for which the Lebesgue outer measure does not give the Lebesgue measure. (Such sets are, in fact, not Lebesgue-measurable.) The Lebesgue measure on R has the following properties: All the above may be succinctly summarized as follows (although the last two assertions are non-trivially linked to the following): The Lebesgue measure also has

1400-638: The Carathéodory criterion , which requires that for every A ⊆ R {\displaystyle A\subseteq \mathbb {R} } , The sets E {\displaystyle E} that satisfy the Carathéodory criterion are said to be Lebesgue-measurable, with its Lebesgue measure being defined as its Lebesgue outer measure: λ ( E ) = λ ∗ ( E ) {\displaystyle \lambda (E)=\lambda ^{\!*\!}(E)} . The set of all such E {\displaystyle E} forms

1450-455: The Cartesian coordinates ( x i , y i ) {\displaystyle (x_{i},y_{i})} ( i =0, 1, ..., n -1) of whose n vertices are known, the area is given by the surveyor's formula : where when i = n -1, then i +1 is expressed as modulus n and so refers to 0. The most basic area formula is the formula for the area of a rectangle . Given

1500-542: The Euclidean metric on R (or any metric Lipschitz equivalent to it). On the other hand, a set may have topological dimension less than n and have positive n -dimensional Lebesgue measure. An example of this is the Smith–Volterra–Cantor set which has topological dimension 0 yet has positive 1-dimensional Lebesgue measure. In order to show that a given set A is Lebesgue-measurable, one usually tries to find

1550-517: The International System of Units (SI), the standard unit of area is the square metre (written as m ), which is 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, and the area of any other shape or surface is a dimensionless real number . There are several well-known formulas for

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1600-403: The greatest lower bound (infimum) of the lengths from among all possible such sets. Intuitively, it is the total length of those interval sets which fit E {\displaystyle E} most tightly and do not overlap. That characterizes the Lebesgue outer measure. Whether this outer measure translates to the Lebesgue measure proper depends on an additional condition. This condition

1650-478: The hectare is still commonly used to measure land: Other uncommon metric units of area include the tetrad , the hectad , and the myriad . The acre is also commonly used to measure land areas, where An acre is approximately 40% of a hectare. On the atomic scale, area is measured in units of barns , such that: The barn is commonly used in describing the cross-sectional area of interaction in nuclear physics . In South Asia (mainly Indians), although

1700-417: The surveyor's formula for the area of any polygon with known vertex locations by Gauss in the 19th century. The development of integral calculus in the late 17th century provided tools that could subsequently be used for computing more complicated areas, such as the area of an ellipse and the surface areas of various curved three-dimensional objects. For a non-self-intersecting ( simple ) polygon,

1750-556: The 5th century BCE, Hippocrates of Chios was the first to show that the area of a disk (the region enclosed by a circle) is proportional to the square of its diameter, as part of his quadrature of the lune of Hippocrates , but did not identify the constant of proportionality . Eudoxus of Cnidus , also in the 5th century BCE, also found that the area of a disk is proportional to its radius squared. Subsequently, Book I of Euclid's Elements dealt with equality of areas between two-dimensional figures. The mathematician Archimedes used

1800-404: The Lebesgue measure on those sets for which it is defined; however, there are many more Lebesgue-measurable sets than there are Borel measurable sets. The Borel measure is translation-invariant, but not complete . The Haar measure can be defined on any locally compact group and is a generalization of the Lebesgue measure ( R with addition is a locally compact group). The Hausdorff measure

1850-466: 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 allowed the development of

1900-501: The area of a triangle is half that of a parallelogram with the same base and height in his book Elements of Geometry . In 499 Aryabhata , a great mathematician - astronomer from the classical age of Indian mathematics and Indian astronomy , expressed the area of a triangle as one-half the base times the height in the Aryabhatiya . In the 7th century CE, Brahmagupta developed a formula, now known as Brahmagupta's formula , for

1950-405: The areas of simple shapes such as triangles , rectangles , 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 usually 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

2000-417: The areas of the approximate parallelograms is exactly π r , which is the area of 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

2050-465: The axiom of choice, non-measurable sets with many surprising properties have been demonstrated, such as those of the Banach–Tarski paradox . In 1970, Robert M. Solovay showed that the existence of sets that are not Lebesgue-measurable is not provable within the framework of Zermelo–Fraenkel set theory in the absence of the axiom of choice (see Solovay's model ). The Borel measure agrees with

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2100-417: The conversion between two square units is the square of the conversion between the corresponding length units. the relationship between square feet and square inches is where 144 = 12 = 12 × 12. Similarly: In addition, conversion factors include: There are several other common units for area. The are was the original unit of area in the metric system , with: Though the are has fallen out of use,

2150-520: The countries use SI units as official, many South Asians still use traditional units. Each administrative division has its own area unit, some of them have same names, but with different values. There's no official consensus about the traditional units values. Thus, the conversions between the SI units and the traditional units may have different results, depending on what reference that has been used. Some traditional South Asian units that have fixed value: In

2200-406: The left. If the triangle is moved to the other side of the trapezoid, then the resulting figure is a rectangle. It follows that the area of the parallelogram is the same as the area of the rectangle: However, the same parallelogram can also be cut along a diagonal into two congruent triangles, as shown in the figure to the right. It follows that the area of each triangle is half the area of

2250-432: The other hand, if geometry is developed before arithmetic , this formula can be used to define multiplication of real numbers . Most other simple formulas for area follow from the method of dissection . This involves cutting a shape into pieces, whose areas must sum to the area of the original shape. For an example, any parallelogram can be subdivided into a trapezoid and a right triangle , as shown in figure to

2300-421: The parallelogram: Similar arguments can be used to find area formulas for the trapezoid as well as more complicated polygons . The formula for the area of a circle (more properly called the area enclosed by a circle or the area of a disk ) is based on a similar method. Given a circle of radius r , it is possible to partition the circle into sectors , as shown in the figure to the right. Each sector

2350-402: The property of being σ -finite . A subset of R is a null set if, for every ε > 0, it can be covered with countably many products of n intervals whose total volume is at most ε. All countable sets are null sets. If a subset of R has Hausdorff dimension less than n then it is a null set with respect to n -dimensional Lebesgue measure. Here Hausdorff dimension is relative to

2400-410: 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 π r , and the area of a triangle is half the base times the height, yielding the area π r for the disk.) Archimedes approximated

2450-511: 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 that such a function exists. An approach to defining what is meant by "area" is through axioms . "Area" can be defined as a function from a collection M of a special kinds of plane figures (termed measurable sets) to the set of real numbers, which satisfies the following properties: It can be proved that such an area function actually exists. Every unit of length has

2500-419: 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 polygons ). Heron of Alexandria found what

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