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49-518: The district has a total land area of 21.98 km. Its administrative center is located at 154 meters above sea level . According to a 2005 estimate by the INEI , the district has 278,804 inhabitants and a population density of 15,736.9 persons/km. In 1999, there were 75,595 households in the district. The high point of Lima's religious calendar for the masses is a month of festivities in October dedicated to

98-439: A cell imposes upper limits on size, as the volume increases much faster than does the surface area, thus limiting the rate at which substances diffuse from the interior across the cell membrane to interstitial spaces or to other cells. Indeed, representing a cell as an idealized sphere of radius r , the volume and surface area are, respectively, V = (4/3) πr and SA = 4 πr . The resulting surface area to volume ratio

147-400: 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 )

196-535: A sphere , are assigned surface area using their representation as parametric surfaces . This definition of surface area is based on methods of infinitesimal calculus and involves partial derivatives and double integration . A general definition of surface area was sought by Henri Lebesgue and Hermann Minkowski at the turn of the twentieth century. Their work led to the development of geometric measure theory , which studies various notions of surface area for irregular objects of any dimension. An important example

245-417: 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

294-512: A given smooth surface. It was demonstrated by Hermann Schwarz that already for the cylinder, different choices of approximating flat surfaces can lead to different limiting values of the area; this example is known as the Schwarz lantern . Various approaches to a general definition of surface area were developed in the late nineteenth and the early twentieth century by Henri Lebesgue and Hermann Minkowski . While for piecewise smooth surfaces there

343-419: A measure of the total area that the surface of the object occupies. The mathematical definition of surface area in the presence of curved surfaces is considerably more involved than the definition of arc length of one-dimensional curves, or of the surface area for polyhedra (i.e., objects with flat polygonal faces ), for which the surface area is the sum of the areas of its faces. Smooth surfaces, such as

392-428: 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

441-450: 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 . Surface area This is an accepted version of this page The surface area (symbol A ) of a solid object is

490-507: Is a unique natural notion of surface area, if a surface is very irregular, or rough, then it may not be possible to assign an area to it at all. A typical example is given by a surface with spikes spread throughout in a dense fashion. Many surfaces of this type occur in the study of fractals . Extensions of the notion of area which partially fulfill its function and may be defined even for very badly irregular surfaces are studied in geometric measure theory . A specific example of such an extension

539-768: Is also home to the city (and country)'s main football stadium , the Estadio Nacional ( National Stadium ). West of the center is the Industrial Area , an industrial belt extending into neighboring Callao Region, and home to the main industries in both city and country. Most of the area is covered by large blocks containing large factories. Also this area include the University City of the National University of San Marcos . At its northern and southern edges, there are clusters of residential areas , particularly in

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588-447: 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

637-410: Is credited to Archimedes . Surface area is important in chemical kinetics . Increasing the surface area of a substance generally increases the rate of a chemical reaction . For example, iron in a fine powder will combust , while in solid blocks it is stable enough to use in structures. For different applications a minimal or maximal surface area may be desired. The surface area of an organism

686-418: 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

735-431: Is defined by the formula Thus the area of S D is obtained by integrating the length of the normal vector r → u × r → v {\displaystyle {\vec {r}}_{u}\times {\vec {r}}_{v}} to the surface over the appropriate region D in the parametric uv plane. The area of the whole surface is then obtained by adding together

784-635: Is important in several considerations, such as regulation of body temperature and digestion . Animals use their teeth to grind food down into smaller particles, increasing the surface area available for digestion. The epithelial tissue lining the digestive tract contains microvilli , greatly increasing the area available for absorption. Elephants have large ears , allowing them to regulate their own body temperature. In other instances, animals will need to minimize surface area; for example, people will fold their arms over their chest when cold to minimize heat loss. The surface area to volume ratio (SA:V) of

833-462: 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

882-744: Is not paved for cars, but almost entirely a shopping and pedestrian street ; the main thoroughfares for cars and buses are Tacna Ave. on the West side and Abancay Ave. on the East. Both are separated from Jirón de la Unión by 4 blocks. The Plaza de Armas (Grand Army Plaza), which is the main square, is located on block 2 of Jirón de la Unión, facing the Peruvian government palace and the Metropolitan Municipality of Lima ( City Hall ). It's also known as Damero de Pizarro ( Pizarro's Checkerboard ). East of

931-431: 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

980-463: Is the Minkowski content of a surface. While the areas of many simple surfaces have been known since antiquity, a rigorous mathematical definition of area requires a great deal of care. This should provide a function which assigns a positive real number to a certain class of surfaces that satisfies several natural requirements. The most fundamental property of the surface area is its additivity :

1029-823: Is the Minkowski content of the surface. r = Internal radius, h = height s = slant height of the cone, r = radius of the circular base, h = height of the cone r → u {\displaystyle {\vec {r}}_{u}} = partial derivative of r → {\displaystyle {\vec {r}}} with respect to u {\displaystyle u} , r → v {\displaystyle {\vec {r}}_{v}} = partial derivative of r → {\displaystyle {\vec {r}}} with respect to v {\displaystyle v} , D {\displaystyle D} = shadow region The below given formulas can be used to show that

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1078-470: 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,

1127-500: 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 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 ,

1176-455: 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

1225-523: The Lord of Miracles , during which take place several processions in the city. Central Lima (known as Cercado proper) is limited by Avenida Alfonso Ugarte on the west and Jirón Huánuco (Huánuco Street) on the east. It is divided into West and East sides by Jirón de la Unión (Union Street), from which cuadras ( blocks ) are numbered beginning at 100 and changing the first numbers at the next block. Unlike New York's Fifth Avenue , though, Jirón de la Unión

1274-478: 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

1323-417: 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,

1372-429: 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 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

1421-556: 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

1470-531: The West Side is Santa Beatriz section, which contains residential buildings and the Parque de la Reserva . Santa Beatriz is locally famous for containing the buildings for the state TV network TNP ( Ch. 7 ), and the top two private TV networks , America Television (Ch. 4) and Panamericana Television (Ch. 5). Its main thoroughfare is Arequipa Avenue , a narrow boulevard lined with trees of all sizes. Santa Beatriz

1519-439: 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 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

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1568-466: 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

1617-501: 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

1666-423: The area of the whole is the sum of the areas of the parts . More rigorously, if a surface S is a union of finitely many pieces S 1 , …, S r which do not overlap except at their boundaries, then Surface areas of flat polygonal shapes must agree with their geometrically defined area . Since surface area is a geometric notion, areas of congruent surfaces must be the same and the area must depend only on

1715-581: 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 usually requires multivariable calculus . Area plays an important role in modern mathematics. In addition to its obvious importance in geometry and calculus, area

1764-417: 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

1813-412: The areas of the pieces, using additivity of surface area. The main formula can be specialized to different classes of surfaces, giving, in particular, formulas for areas of graphs z = f ( x , y ) and surfaces of revolution . One of the subtleties of surface area, as compared to arc length of curves, is that surface area cannot be defined simply as the limit of areas of polyhedral shapes approximating

1862-572: The center is the Barrios Altos (Uptown) neighborhood. Here the oldest, though least stable, buildings in Central Lima are located. Two cemeteries, El Ángel and Presbítero Maestro , form the eastern border with El Agustino . Parts of the long-demolished colonial city walls can be seen here. Abutting this to the southwest is the Barrio chino (Chinatown) neighborhood, dating from the mid-1800s. South of

1911-417: 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,

1960-520: 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

2009-406: 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

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2058-432: 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

2107-421: 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

2156-536: The shape of the surface, but not on its position and orientation in space. This means that surface area is invariant under the group of Euclidean motions . These properties uniquely characterize surface area for a wide class of geometric surfaces called piecewise smooth . Such surfaces consist of finitely many pieces that can be represented in the parametric form with a continuously differentiable function r → . {\displaystyle {\vec {r}}.} The area of an individual piece

2205-415: The southern zone bordering Pueblo Libre , San Miguel and Callao Region's Bellavista District . Area Area is the measure of a region 's size on a surface . The area of 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

2254-865: The surface area of a sphere and cylinder of the same radius and height are in the ratio 2 : 3 , as follows. Let the radius be r and the height be h (which is 2 r for the sphere). Sphere surface area = 4 π r 2 = ( 2 π r 2 ) × 2 Cylinder surface area = 2 π r ( h + r ) = 2 π r ( 2 r + r ) = ( 2 π r 2 ) × 3 {\displaystyle {\begin{array}{rlll}{\text{Sphere surface area}}&=4\pi r^{2}&&=(2\pi r^{2})\times 2\\{\text{Cylinder surface area}}&=2\pi r(h+r)&=2\pi r(2r+r)&=(2\pi r^{2})\times 3\end{array}}} The discovery of this ratio

2303-410: 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

2352-511: 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

2401-419: 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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