Sphere
A sphere is a round geometrical object in three-dimensional space, the surface of a round ball. Like a circle in a two-dimensional space, a sphere is defined mathematically as the set of points that are all at the same distance r from a given point, but in a three-dimensional space; this distance r is the radius of the ball, made up from all points with a distance less than r from the given point, the center of the mathematical ball. These are referred to as the radius and center of the sphere, respectively; the longest straight line segment through the ball, connecting two points of the sphere, passes through the center and its length is thus twice the radius. While outside mathematics the terms "sphere" and "ball" are sometimes used interchangeably, in mathematics the above distinction is made between a sphere, a two-dimensional closed surface, embedded in a three-dimensional Euclidean space, a ball, a three-dimensional shape that includes the sphere and everything inside the sphere, or, more just the points inside, but not on the sphere.
The distinction between ball and sphere has not always been maintained and older mathematical references talk about a sphere as a solid. This is analogous to the situation in the plane, where the terms "circle" and "disk" can be confounded. In analytic geometry, a sphere with center and radius r is the locus of all points such that 2 + 2 + 2 = r 2. Let a, b, c, d, e be real numbers with a ≠ 0 and put x 0 = − b a, y 0 = − c a, z 0 = − d a, ρ = b 2 + c 2 + d 2 − a e a 2; the equation f = a + 2 + e = 0 has no real points as solutions if ρ < 0 and is called the equation of an imaginary sphere. If ρ = 0 the only solution of f = 0 is the point P 0 = and the equation is said to be the equation of a point sphere. In the case ρ > 0, f = 0 is an equation of a sphere whose center is P 0 and whose radius is ρ. If a in the above equation is zero f = 0 is the equation of a plane. Thus, a plane may be thought of as a sphere of infinite radius; the points on the sphere with radius r > 0 and center can be parameterized via x = x 0 + r sin θ cos φ y = y 0 + r sin θ sin φ z = z 0 + r cos θ The parameter θ {
Area
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
Regular polygon
In Euclidean geometry, a regular polygon is a polygon, equiangular and equilateral. Regular polygons may be either star. In the limit, a sequence of regular polygons with an increasing number of sides approximates a circle, if the perimeter or area is fixed, or a regular apeirogon, if the edge length is fixed; these properties apply to all regular polygons, whether star. A regular n-sided polygon has rotational symmetry of order n. All vertices of a regular polygon lie on a common circle; that is, a regular polygon is a cyclic polygon. Together with the property of equal-length sides, this implies that every regular polygon has an inscribed circle or incircle, tangent to every side at the midpoint, thus a regular polygon is a tangential polygon. A regular n-sided polygon can be constructed with compass and straightedge if and only if the odd prime factors of n are distinct Fermat primes. See constructible polygon; the symmetry group of an n-sided regular polygon is dihedral group Dn: D2, D3, D4...
It consists of the rotations in Cn, together with reflection symmetry in n axes that pass through the center. If n is then half of these axes pass through two opposite vertices, the other half through the midpoint of opposite sides. If n is odd all axes pass through a vertex and the midpoint of the opposite side. All regular simple polygons are convex; those having the same number of sides are similar. An n-sided convex regular polygon is denoted by its Schläfli symbol. For n < 3, we have two degenerate cases: Monogon Degenerate in ordinary space. Digon. In certain contexts all the polygons considered. In such circumstances it is customary to drop the prefix regular. For instance, all the faces of uniform polyhedra must be regular and the faces will be described as triangle, pentagon, etc. For a regular convex n-gon, each interior angle has a measure of: × 180 degrees, or equivalently 180 n degrees; as the number of sides, n approaches infinity, the internal angle approaches 180 degrees. For a regular polygon with 10,000 sides the internal angle is 179.964°.
As the number of sides increase, the internal angle can come close to 180°, the shape of the polygon approaches that of a circle. However the polygon can never become a circle; the value of the internal angle can never become equal to 180°, as the circumference would become a straight line. For this reason, a circle is not a polygon with an infinite number of sides. For n > 2, the number of diagonals is 1 2 n. The diagonals divide the polygon into 1, 4, 11, 24, … pieces OEIS: A007678. For a regular n-gon inscribed in a unit-radius circle, the product of the distances from a given vertex to all other vertices equals n. For a regular simple n-gon with circumradius R and distances di from an arbitrary point in the plane to the vertices, we have ∑ i = 1 n d i 4 n + 3 R 4 = 2. For a regular n-gon, the sum of the perpendicular distances from any interior point to the n sides is n times the apothem; this is a generalization of Viviani's theorem for the n. The circumradius R from the center of a regular polygon to one of the vertices is related to the side length s or to the apothem a by R = s 2 sin = a cos
Collinearity
In geometry, collinearity of a set of points is the property of their lying on a single line. A set of points with this property is said to be collinear. In greater generality, the term has been used for aligned objects, that is, things being "in a line" or "in a row". In any geometry, the set of points on a line are said to be collinear. In Euclidean geometry this relation is intuitively visualized by points lying in a row on a "straight line". However, in most geometries a line is a primitive object type, so such visualizations will not be appropriate. A model for the geometry offers an interpretation of how the points and other object types relate to one another and a notion such as collinearity must be interpreted within the context of that model. For instance, in spherical geometry, where lines are represented in the standard model by great circles of a sphere, sets of collinear points lie on the same great circle; such points do not lie on a "straight line" in the Euclidean sense, are not thought of as being in a row.
A mapping of a geometry to itself which sends lines to lines is called a collineation. The linear maps of vector spaces, viewed as geometric maps, map lines to lines. In projective geometry these linear mappings are called homographies and are just one type of collineation. In any triangle the following sets of points are collinear: The orthocenter, the circumcenter, the centroid, the Exeter point, the de Longchamps point, the center of the nine-point circle are collinear, all falling on a line called the Euler line; the de Longchamps point has other collinearities. Any vertex, the tangency of the opposite side with an excircle, the Nagel point are collinear in a line called a splitter of the triangle; the midpoint of any side, the point, equidistant from it along the triangle's boundary in either direction, the center of the Spieker circle are collinear in a line called a cleaver of the triangle. Any vertex, the tangency of the opposite side with the incircle, the Gergonne point are collinear.
From any point on the circumcircle of a triangle, the nearest points on each of the three extended sides of the triangle are collinear in the Simson line of the point on the circumcircle. The lines connecting the feet of the altitudes intersect the opposite sides at collinear points. A triangle's incenter, the midpoint of an altitude, the point of contact of the corresponding side with the excircle relative to that side are collinear. Menelaus' theorem states that three points P 1, P 2, P 3 on the sides of a triangle opposite vertices A 1, A 2, A 3 are collinear if and only if the following products of segment lengths are equal: P 1 A 2 ⋅ P 2 A 3 ⋅ P 3 A 1 = P 1 A 3 ⋅ P 2 A 1 ⋅ P 3 A 2; the incenter, the centroid, the Spieker circle's center are collinear. The circumcenter, the Brocard midpoint, the Lemoine point of a triangle are collinear. Two perpendicular lines intersecting at the orthocenter of a triangle each intersect each of the triangle's extended sides; the midpoints on the three sides of these points of intersection are collinear in the Droz–Farny line.
In a convex quadrilateral ABCD whose opposite sides intersect at E and F, the midpoints of AC, BD, EF are collinear and the line through them is called the Newton line. If the quadrilateral is a tangential quadrilateral its incenter lies on this line. In a convex quadrilateral, the quasiorthocenter H, the "area centroid" G, the quasicircumcenter O are collinear in this order, HG = 2GO. Other collinearities of a tangential quadrilateral are given in Tangential quadrilateral#Collinear points. In a cyclic quadrilateral, the circumcenter, the vertex centroid, the anticenter are collinear. In a cyclic quadrilateral, the area centroid, the vertex centroid, the intersection of the diagonals are collinear. In a tangential trapezoid, the tangencies of the incircle with the two bases are collinear with the incenter. In a tangential trapezoid, the midpoints of the legs are collinear with the incenter. Pascal's theorem states that if an arbitrary six points are chosen on a conic section and joined by line segments in any order to form a hexagon the three pairs of opposite sides of the hexagon meet in three points which lie on a straight line, called the Pascal line of the hexagon.
The converse is true: the Braikenridge–Maclaurin theorem states that if the three intersection points of the three pairs of lines through opposite sides of a hexagon lie on a line the six ver
Law of sines
In trigonometry, the law of sines, sine law, sine formula, or sine rule is an equation relating the lengths of the sides of a triangle to the sines of its angles. According to the law, a sin A = b sin B = c sin C = d, where a, b, c are the lengths of the sides of a triangle, A, B, C are the opposite angles, while d is the diameter of the triangle's circumcircle; when the last part of the equation is not used, the law is sometimes stated using the reciprocals. The law of sines can be used to compute the remaining sides of a triangle when two angles and a side are known—a technique known as triangulation, it can be used when two sides and one of the non-enclosed angles are known. In some such cases, the triangle is not uniquely determined by this data and the technique gives two possible values for the enclosed angle; the law of sines is one of two trigonometric equations applied to find lengths and angles in scalene triangles, with the other being the law of cosines. The law of sines can be generalized to higher dimensions on surfaces with constant curvature.
The area T of any triangle can be written as one half of its base times its height. Selecting one side of the triangle as the base, the height of the triangle relative to that base is computed as the length of another side times the sine of the angle between the chosen side and the base. Thus, depending on the selection of the base the area of the triangle can be written as any of: T = 1 2 b = 1 2 c = 1 2 a. Multiplying these by 2/abc gives 2 T a b c = sin A a = sin B b = sin C c; when using the law of sines to find a side of a triangle, an ambiguous case occurs when two separate triangles can be constructed from the data provided. In the case shown below they are triangles ABC and AB′C′. Given a general triangle, the following conditions would need to be fulfilled for the case to be ambiguous: The only information known about the triangle is the angle A and the sides a and c; the angle A is acute. The side a is shorter than the side c; the side a is longer than the altitude h from angle B, where h = c sin A.
If all the above conditions are true each of angles C and C′ produces a valid triangle, meaning that both of the following are true: C ′ = arcsin c sin A a or C = π − arcsin c sin A a. From there we can find the corresponding B and b or B′ and b′ if required, where b is the side bounded by angles A and C and b′ bounded by A and C′. Without further information it is impossible to decide, the triangle being asked for; the following are examples of. Given: side a = 20, side c = 24, angle C = 40°. Angle A is desired. Using the law of sines, we conclude that sin A 20 = sin 40 ∘ 24. A = arcsin ≈ 32.39 ∘. Note that the potential solution A = 147.61° is excluded because that would give A + B + C > 180°. If the lengths of two sides of the triangle a and b are equal to x, the third side has length c, the angles opposite the sides of lengths a, b, c are A, B, C then A = B = 180 ∘ − C 2 = 90 ∘ − C 2 sin A =
Circle
A circle is a simple closed shape. It is the set of all points in a plane; the distance between any of the points and the centre is called the radius. This article is about circles in Euclidean geometry, and, in particular, the Euclidean plane, except where otherwise noted. A circle is a simple closed curve that divides the plane into two regions: an interior and an exterior. In everyday use, the term "circle" may be used interchangeably to refer to either the boundary of the figure, or to the whole figure including its interior. A circle may be defined as a special kind of ellipse in which the two foci are coincident and the eccentricity is 0, or the two-dimensional shape enclosing the most area per unit perimeter squared, using calculus of variations. A circle is a plane figure bounded by one line, such that all right lines drawn from a certain point within it to the bounding line, are equal; the bounding line is called the point, its centre. Annulus: a ring-shaped object, the region bounded by two concentric circles.
Arc: any connected part of a circle. Specifying two end points of an arc and a center allows for two arcs that together make up a full circle. Centre: the point equidistant from all points on the circle. Chord: a line segment whose endpoints lie on the circle, thus dividing a circle in two sements. Circumference: the length of one circuit along the circle, or the distance around the circle. Diameter: a line segment whose endpoints lie on the circle and that passes through the centre; this is the largest distance between any two points on the circle. It is a special case of a chord, namely the longest chord for a given circle, its length is twice the length of a radius. Disc: the region of the plane bounded by a circle. Lens: the region common to two overlapping discs. Passant: a coplanar straight line that has no point in common with the circle. Radius: a line segment joining the centre of a circle with any single point on the circle itself. Sector: a region bounded by two radii of equal length with a common center and either of the two possible arcs, determined by this center and the endpoints of the radii.
Segment: a region bounded by a chord and one of the arcs connecting the chord's endpoints. The length of the chord imposes a lower boundary on the diameter of possible arcs. Sometimes the term segment is used only for regions not containing the center of the circle to which their arc belongs to. Secant: an extended chord, a coplanar straight line, intersecting a circle in two points. Semicircle: one of the two possible arcs determined by the endpoints of a diameter, taking its midpoint as center. In non-technical common usage it may mean the interior of the two dimensional region bounded by a diameter and one of its arcs, technically called a half-disc. A half-disc is a special case of a segment, namely the largest one. Tangent: a coplanar straight line that has one single point in common with a circle. All of the specified regions may be considered as open, that is, not containing their boundaries, or as closed, including their respective boundaries; the word circle derives from the Greek κίρκος/κύκλος, itself a metathesis of the Homeric Greek κρίκος, meaning "hoop" or "ring".
The origins of the words circus and circuit are related. The circle has been known since before the beginning of recorded history. Natural circles would have been observed, such as the Moon, a short plant stalk blowing in the wind on sand, which forms a circle shape in the sand; the circle is the basis for the wheel, with related inventions such as gears, makes much of modern machinery possible. In mathematics, the study of the circle has helped inspire the development of geometry and calculus. Early science geometry and astrology and astronomy, was connected to the divine for most medieval scholars, many believed that there was something intrinsically "divine" or "perfect" that could be found in circles; some highlights in the history of the circle are: 1700 BCE – The Rhind papyrus gives a method to find the area of a circular field. The result corresponds to 256/81 as an approximate value of π. 300 BCE – Book 3 of Euclid's Elements deals with the properties of circles. In Plato's Seventh Letter there is a detailed explanation of the circle.
Plato explains the perfect circle, how it is different from any drawing, definition or explanation. 1880 CE – Lindemann proves that π is transcendental settling the millennia-old problem of squaring the circle. The ratio of a circle's circumference to its diameter is π, an irrational constant equal to 3.141592654. Thus the length of the circumference C is related to the radius r and diameter d by: C = 2 π r = π d; as proved by Archimedes, in his Measurement of a Circle, the area enclosed by a circle is equal to that of a triangle whose base has the length of the circle's circumference and whose height equals the circle's radius, which comes to π multiplied by the radius squared: A r e a = π r 2. Equivalently, denoting diameter by d, A r e
International Standard Serial Number
An International Standard Serial Number is an eight-digit serial number used to uniquely identify a serial publication, such as a magazine. The ISSN is helpful in distinguishing between serials with the same title. ISSN are used in ordering, interlibrary loans, other practices in connection with serial literature; the ISSN system was first drafted as an International Organization for Standardization international standard in 1971 and published as ISO 3297 in 1975. ISO subcommittee TC 46/SC 9 is responsible for maintaining the standard; when a serial with the same content is published in more than one media type, a different ISSN is assigned to each media type. For example, many serials are published both in electronic media; the ISSN system refers to these types as electronic ISSN, respectively. Conversely, as defined in ISO 3297:2007, every serial in the ISSN system is assigned a linking ISSN the same as the ISSN assigned to the serial in its first published medium, which links together all ISSNs assigned to the serial in every medium.
The format of the ISSN is an eight digit code, divided by a hyphen into two four-digit numbers. As an integer number, it can be represented by the first seven digits; the last code digit, which may be 0-9 or an X, is a check digit. Formally, the general form of the ISSN code can be expressed as follows: NNNN-NNNC where N is in the set, a digit character, C is in; the ISSN of the journal Hearing Research, for example, is 0378-5955, where the final 5 is the check digit, C=5. To calculate the check digit, the following algorithm may be used: Calculate the sum of the first seven digits of the ISSN multiplied by its position in the number, counting from the right—that is, 8, 7, 6, 5, 4, 3, 2, respectively: 0 ⋅ 8 + 3 ⋅ 7 + 7 ⋅ 6 + 8 ⋅ 5 + 5 ⋅ 4 + 9 ⋅ 3 + 5 ⋅ 2 = 0 + 21 + 42 + 40 + 20 + 27 + 10 = 160 The modulus 11 of this sum is calculated. For calculations, an upper case X in the check digit position indicates a check digit of 10. To confirm the check digit, calculate the sum of all eight digits of the ISSN multiplied by its position in the number, counting from the right.
The modulus 11 of the sum must be 0. There is an online ISSN checker. ISSN codes are assigned by a network of ISSN National Centres located at national libraries and coordinated by the ISSN International Centre based in Paris; the International Centre is an intergovernmental organization created in 1974 through an agreement between UNESCO and the French government. The International Centre maintains a database of all ISSNs assigned worldwide, the ISDS Register otherwise known as the ISSN Register. At the end of 2016, the ISSN Register contained records for 1,943,572 items. ISSN and ISBN codes are similar in concept. An ISBN might be assigned for particular issues of a serial, in addition to the ISSN code for the serial as a whole. An ISSN, unlike the ISBN code, is an anonymous identifier associated with a serial title, containing no information as to the publisher or its location. For this reason a new ISSN is assigned to a serial each time it undergoes a major title change. Since the ISSN applies to an entire serial a new identifier, the Serial Item and Contribution Identifier, was built on top of it to allow references to specific volumes, articles, or other identifiable components.
Separate ISSNs are needed for serials in different media. Thus, the print and electronic media versions of a serial need separate ISSNs. A CD-ROM version and a web version of a serial require different ISSNs since two different media are involved. However, the same ISSN can be used for different file formats of the same online serial; this "media-oriented identification" of serials made sense in the 1970s. In the 1990s and onward, with personal computers, better screens, the Web, it makes sense to consider only content, independent of media; this "content-oriented identification" of serials was a repressed demand during a decade, but no ISSN update or initiative occurred. A natural extension for ISSN, the unique-identification of the articles in the serials, was the main demand application. An alternative serials' contents model arrived with the indecs Content Model and its application, the digital object identifier, as ISSN-independent initiative, consolidated in the 2000s. Only in 2007, ISSN-L was defined in the