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48-401: When used in weather charts, okta measurements are shown by means of graphic symbols (rather than numerals) contained within weather circles, to which are attached further symbols indicating other measured data such as wind speed and wind direction. Although relatively straightforward to measure (visually, for instance, by using a mirror), oktas only estimate cloud cover in terms of the area of

96-480: A chemical formula . The informal use of the term formula in science refers to the general construct of a relationship between given quantities . The plural of formula can be either formulas (from the most common English plural noun form ) or, under the influence of scientific Latin , formulae (from the original Latin ). In mathematics , a formula generally refers to an equation or inequality relating one mathematical expression to another, with

144-400: 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 )

192-436: A sine curve to model the movement of the tides in a bay , may be created to solve a particular problem. In all cases, however, formulas form the basis for calculations. Expressions are distinct from formulas in the sense that they don't usually contain relations like equality (=) or inequality (<). Expressions denote a mathematical object , where as formulas denote a statement about mathematical objects. This

240-417: 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

288-407: A general context, formulas often represent mathematical models of real world phenomena, and as such can be used to provide solutions (or approximate solutions) to real world problems, with some being more general than others. For example, the formula is an expression of Newton's second law , and is applicable to a wide range of physical situations. Other formulas, such as the use of the equation of

336-432: 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 is used to refer to the region, as in a " polygonal area ". The area of

384-584: A modified version of the Baudot-Murray code . There are some symbols in Unicode which resemble those used for oktas. However, some okta symbols lack a similar-looking Unicode character. The use of Unicode to render oktas depends on whether a font with these characters is installed; Unicode symbols may be difficult to work with in software that does not render these characters uniformly. The Unicode set of related symbols includes: This cloud –related article

432-447: A net negative charge . A chemical formula identifies each constituent element by its chemical symbol , and indicates the proportionate number of atoms of each element. In empirical formulas , these proportions begin with a key element and then assign numbers of atoms of the other elements in the compound—as ratios to the key element. For molecular compounds, these ratio numbers can always be expressed as whole numbers. For example,

480-428: 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

528-477: 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 , the unit square is defined to have area one, and

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576-458: 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 . Formula In science , a formula is a concise way of expressing information symbolically, as in a mathematical formula or

624-450: 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 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

672-516: Is a way of expressing information about the proportions of atoms that constitute a particular chemical compound , using a single line of chemical element symbols , numbers , and sometimes other symbols, such as parentheses, brackets, and plus (+) and minus (−) signs. For example, H 2 O is the chemical formula for water , specifying that each molecule consists of two hydrogen (H) atoms and one oxygen (O) atom. Similarly, O 3 denotes an ozone molecule consisting of three oxygen atoms and

720-407: Is always stored in the cell itself, making the stating of the name redundant. Formulas used in science almost always require a choice of units. Formulas are used to express relationships between various quantities, such as temperature, mass, or charge in physics; supply, profit, or demand in economics; or a wide range of other quantities in other disciplines. An example of a formula used in science

768-511: Is analogous to natural language, where a noun phrase refers to an object, and a whole sentence refers to a fact. For example, 8 x − 5 {\displaystyle 8x-5} is an expression, while 8 x − 5 ≥ 3 {\displaystyle 8x-5\geq 3} is a formula. However, in some areas mathematics, and in particular in computer algebra , formulas are viewed as expressions that can be evaluated to true or false , depending on

816-447: 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

864-418: 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

912-462: 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

960-405: Is often implicitly provided in the form of a computer instruction such as. In computer spreadsheet software, a formula indicating how to compute the value of a cell , say A3 , could be written as where A1 and A2 refer to other cells (column A, row 1 or 2) within the spreadsheet. This is a shortcut for the "paper" form A3 = A1+A2 , where A3 is, by convention, omitted because the result

1008-431: 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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1056-470: 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,

1104-455: 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

1152-478: 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

1200-417: 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,

1248-556: 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

1296-466: 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

1344-501: 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

1392-424: 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 the area. Indeed, the problem of determining

1440-535: 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

1488-417: 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

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1536-410: The chemical compound of the formula consists of simple molecules , chemical formulas often employ ways to suggest the structure of the molecule. There are several types of these formulas, including molecular formulas and condensed formulas . A molecular formula enumerates the number of atoms to reflect those in the molecule, so that the molecular formula for glucose is C 6 H 12 O 6 rather than

1584-417: 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,

1632-520: 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

1680-556: The early 20th century, it was common for weather maps to be hand drawn. The symbols for cloud cover on these maps, like the modern symbols, were drawn inside the circle marking the position of the weather station making the measurements. Unlike the modern symbols, these ones consisted of straight lines only rather than filled in blocks which would have been less practical on a hand drawing. A reduced set of these symbols were used on teleprinters used for distributing weather information and warnings. These machines were 5-bit teleprinters using

1728-434: The empirical formula of ethanol may be written C 2 H 6 O, because the molecules of ethanol all contain two carbon atoms, six hydrogen atoms, and one oxygen atom. Some types of ionic compounds, however, cannot be written as empirical formulas which contains only the whole numbers. An example is boron carbide , whose formula of CB n is a variable non-whole number ratio, with n ranging from over 4 to more than 6.5. When

1776-444: The glucose empirical formula, which is CH 2 O. Except for the very simple substances, molecular chemical formulas generally lack needed structural information, and might even be ambiguous in occasions. A structural formula is a drawing that shows the location of each atom, and which atoms it binds to. In computing , a formula typically describes a calculation , such as addition, to be performed on one or more variables. A formula

1824-406: 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

1872-436: The most important ones being mathematical theorems . For example, determining the volume of a sphere requires a significant amount of integral calculus or its geometrical analogue, the method of exhaustion . However, having done this once in terms of some parameter (the radius for example), mathematicians have produced a formula to describe the volume of a sphere in terms of its radius: Having obtained this result,

1920-432: 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

1968-421: 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

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2016-715: The sky covered by clouds. They do not account for cloud type or thickness, and this limits their use for estimating cloud albedo or surface solar radiation receipt. Cloud oktas can also be measured using satellite imagery from geostationary satellites equipped with high-resolution image sensors such as Himawari-8 . Similar to traditional approaches, satellite images do not account for cloud composition. Oktas are often referenced in aviation weather forecasts and low level forecasts: SKC = Sky clear (0 oktas); FEW = Few (1 to 2 oktas); SCT = Scattered (3 to 4 oktas); BKN = Broken (5 to 7 oktas); OVC = Overcast (8 oktas); NSC = nil significant cloud; CAVOK = ceiling and visibility okay. In

2064-399: The symbols and formation rules of a given logical language . For example, in first-order logic , is a formula, provided that f {\displaystyle f} is a unary function symbol, P {\displaystyle P} a unary predicate symbol, and Q {\displaystyle Q} a ternary predicate symbol. In modern chemistry , a chemical formula

2112-410: 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

2160-511: 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

2208-419: 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

2256-417: The values that are given to the variables occurring in the expressions. For example 8 x − 5 ≥ 3 {\displaystyle 8x-5\geq 3} takes the value false if x is given a value less than 1, and the value true otherwise. (See Boolean expression ) In mathematical logic , a formula (often referred to as a well-formed formula ) is an entity constructed using

2304-439: The volume of any sphere can be computed as long as its radius is known. Here, notice that the volume V and the radius r are expressed as single letters instead of words or phrases. This convention, while less important in a relatively simple formula, means that mathematicians can more quickly manipulate formulas which are larger and more complex. Mathematical formulas are often algebraic , analytical or in closed form . In

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