Area is a quantity Quantity is a kind of property which exists as magnitude or multitude. It is among the basic classes of things along with quality, substance, change, and relation. Quantity was first introduced as quantum, an entity having quantity. Being a fundamental term, quantity is used to refer to any type of quantitative properties or attributes of things expressing the two-dimensional In mathematics and physics, the dimension of a space or object is informally defined as the minimum number of coordinates needed to specify each point within it. Thus a line has a dimension of one because only one coordinate is needed to specify a point on it. A surface such as a plane or the surface of a cylinder or sphere has a dimension of two size of a defined part of a surface In mathematics, specifically in topology, a surface is a two-dimensional topological manifold. The most familiar examples are those that arise as the boundaries of solid objects in ordinary three-dimensional Euclidean space R3 — for example, the surface of a ball. On the other hand, there are surfaces, such as the Klein bottle, that cannot be, typically a region bounded by a closed curve In mathematics, a curve is, generally speaking, an object similar to a straight line but which is not required to be straight. Often curves in two-dimensional or three-dimensional (space curves) Euclidean space are of interest. The surface area Surface area is the measure of how much exposed area a solid object has, expressed in square units. Mathematical description of the surface area is considerably more involved than the definition of arc length of a curve. For polyhedra the surface area is the sum of the areas of its faces. Smooth surfaces, such as a sphere, are assigned surface of a 3-dimensional solid is the total area of the exposed surface, such as the sum of the areas of the exposed sides of a polyhedron A polyhedron is a geometric solid in three dimensions with flat faces and straight edges. The word polyhedron comes from the Classical Greek πολύεδρον, as poly- (stem of πολύς, "many") + -edron (form of έδρα, "base", "seat", or "face"). Area is an important invariant The most fundamental example of invariance is expressed in our ability to count. For a finite collection of objects of any kind, there appears to be a number to which we invariably arrive regardless of how we count the objects in the set. The quantity – a cardinal number – is associated with the set and is invariant under the process of in the differential geometry of surfaces In mathematics, the differential geometry of surfaces deals with smooth surfaces with various additional structures, most often, a Riemannian metric. Surfaces have been extensively studied from various perspectives: extrinsically, relating to their embedding in Euclidean space and intrinsically, reflecting their properties determined solely by the.[1]
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Units
Units for measuring area, with exact conversions, include:
- square metre The square metre is the SI derived unit of area, with symbol m2 (33A1 in Unicode). It is defined as the area of a square whose sides measure exactly one metre. The square metre is derived from the SI base unit of the metre, which in turn is defined as the length of the path travelled by light in absolute vacuum during a time interval of 1⁄299,792 (m2)
- are The hectare is a unit of area, defined as 10,000 square metres, and primarily used in the measurement of land. In 1795, when the metric system was introduced, the are was defined as being 100 square metres and the hectare was thus 100 ares. When the metric system was rationalised in 1960 with the introduction of the International System of Units ( (a) = 100 square metres (m2)
- hectare The hectare is a unit of area, defined as 10,000 square metres, and primarily used in the measurement of land. In 1795, when the metric system was introduced, the are was defined as being 100 square metres and the hectare was thus 100 ares or 1/100 km2. When the metric system was rationalised in 1960 with the introduction of the International (ha) = 100 ares = 10000 square metres
- square kilometre Square kilometre, symbol km2, is a decimal multiple of the SI unit of surface area, the square metre, one of the SI derived units. 1 km2 is equal to: (km2) = 100 hectares = 10000 ares = 1000000 square metres
- square megametre mega is an SI prefix in the SI system of units denoting a factor of 106, 1,000,000 (one million) (Mm2) = 1000000000000 square metres
- square foot The square foot is an imperial unit / U.S. customary unit of area, used mainly in the United States, the United Kingdom, Hong Kong, Japan, Afghanistan and Canada. It is defined as the area of a square with sides of 1 foot (0.333... yards, 12 inches, or 0.3048 metres) in length. For comparison, the average one-family house in the United States in 20 = 144 square inches = 0.09290304 square metres
- square yard The square yard is an imperial/US customary unit of area, formerly used in most of the English-speaking world but now generally replaced by the square metre outside of the US. It is defined as the area of a square with sides of one yard (three feet, thirty-six inches, 0.9144 metres) in length = 9 square feet = 0.83612736 square metres
- square perch = 30.25 square yards = 25.2928526 square metres
- acre The acre is a unit of area in a number of different systems, including the imperial and U.S. customary systems. The most commonly used acres today are the international acre and, in the United States, the survey acre. The most common use of the acre is to measure tracts of land = 10 square chains A chain is a unit of length; it measures 66 feet or 22 yards or 4 rods or 100 links . There are 10 chains in a furlong, and 80 chains in one statute mile. An acre is the area of 10 square chains (that is, an area of one chain by one furlong). The chain has been used for several centuries in Britain and in some other countries influenced by British = one furlong A furlong is a measure of distance in imperial units and U.S. customary units. It is equal to one-eighth of an international mile, to 220 yards, and to 660 feet. Since furlongs are not used for precision measurements, there is no need to consider other slightly different conversions by one chain = 160 square perches = 4840 square yards = 43560 square feet = 4046.8564224 square metres
- square mile The square mile is an imperial and US unit of measure for an area equal to the area of a square of one statute mile. It should not be confused with miles square, which refers to the number of miles on each side squared. For instance, 20 miles square (20 × 20 miles) is equal to 400 square miles = 640 acres = 2.589988110336 square kilometres
- Kanal (unit) = 0.125 acres.
Formulae
The above calculations show how to find the area of many common shapes.
The area of irregular polygons can be calculated using the "Surveyor's formula".[2]
Additional formulae
Areas of 2-dimensional figures
- a triangle: (where B is any side, and h is the distance from the line on which B lies to the other vertex of the triangle). This formula can be used if the height h is known. If the lengths of the three sides are known then Heron's formula can be used: (where a, b, c are the sides of the triangle, and is half of its perimeter) If an angle and its two included sides are given, then area=absinC where C is the given angle and a and b are its included sides. If the triangle is graphed on a coordinate plane, a matrix can be used and is simplified to the absolute value of (x1y2+ x2y3+ x3y1 - x2y1- x3y2- x1y3) all divided by 2. This formula is also known as the shoelace formula and is an easy way to solve for the area of a coordinate triangle by substituting the 3 points, (x1,y1) (x2,y2) (x3,y 3). The shoelace formula can also be used to find the areas of other polygons when their vertices are known. Another approach for a coordinate triangle is to use Infinitesimal calculus to find the area.
- a simple polygon constructed on a grid of equal-distanced points (i.e., points with integer coordinates) such that all the polygon's vertices are grid points: , where i is the number of grid points inside the polygon and b is the number of boundary points. This result is known as Pick's theorem.
Area in calculus
The area between two graphs can be evaluated by calculating the difference between the integrals of the two functions- the area between the graphs of two functions is equal to the integral of one function, f(x), minus the integral of the other function, g(x).
- an area bounded by a function r = r(θ) expressed in polar coordinates is .
- the area enclosed by a parametric curve with endpoints is given by the line integrals
(see Green's theorem)
- or the z-component of
Surface area of 3-dimensional figures
- cube: 6s2, where s is the length of the top side
- rectangular box: the length divided by height
- cone: , where r is the radius of the circular base, and h is the height. That can also be rewritten as πr2 + πrl where r is the radius and l is the slant height of the cone. πr2 is the base area while πrl is the lateral surface area of the cone.
- prism: 2 × Area of Base + Perimeter of Base × Height
General formula
The general formula for the surface area of the graph of a continuously differentiable function z = f(x,y), where and D is a region in the xy-plane with the smooth boundary:
Even more general formula for the area of the graph of a parametric surface in the vector form where is a continuously differentiable vector function of :
Area minimisation
Given a wire contour, the surface of least area spanning ("filling") it is a minimal surface. Familiar examples include soap bubbles.
The question of the filling area of the Riemannian circle remains open.
See also
- Equi-areal mapping
- Integral
- Orders of magnitude (area)—A list of areas by size.
- Volume
References
Notes
- ^ a b do Carmo, Manfredo. Differential Geometry of Curves and Surfaces. Prentice-Hall, 1976. Page 98.
- ^ http://www.maa.org/pubs/Calc_articles/ma063.pdf
External links
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Q. A square is divided into four congruent regions. One-third of the area of the first region is removed, one-sixth of the area of the second region is removed, and one-twelfth of the area of the third region is removed.
Asked by Kwanna R - Tue Oct 9 00:53:03 2007 - - 2 Answers - 0 Comments
A. 1/3 * 1/4 + 1/6 * 1/4 + 1/12*1/4 = 7/48. 7/48 has been removed.
Answered by C C - Tue Oct 9 00:58:32 2007
