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area

(âr'ē-ə)

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area

To calculate the area of a rectangle, multiply the length by the width. The area of this rectangle is 50 square feet.
(Academy Artworks)

A roughly bounded part of the space on a surface; a region: a farming area; the New York area.
A surface, especially an open, unoccupied piece of ground: a landing area; a playing area.
A distinct part or section, as of a building, set aside for a specific function: a storage area in the basement.
A division of experience, activity, or knowledge; a field: studies in the area of finance; a job in the health-care area.
An open, sunken space next to a building; an areaway.
(Abbr. A) The extent of a planar region or of the surface of a solid measured in square units.
Computer Science. A section of storage set aside for a particular purpose.

[Latin ārea, open space; possibly akin to ārēre, to be dry. See arid.]

areal ar'e·al adj.
areally ar'e·al·ly adv.

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Sci-Tech Encyclopedia: Area

The superficial contents of a geometrical figure of two dimensions. The area of any rectangle or square is the product of two adjacent sides, one of which may be called the base and the other the altitude. In general, any line segment that partially bounds a plane geometric figure may be called a base if its line does not separate the figure, and a perpendicular drawn to the base line from one of its points at greatest distance may be called the altitude. Some area formulas are given in the table. See also Euclidean geometry.

Area formulas
Figure	Formula
Triangle	${hb}\over{2}$ , where h = altitude, b = base; $\sqrt{s(s- a)(s-b)(s-c)}$ , where s = ½(a + b + c), and a, b, and c are sides of the triangle
Rectangle	ab, where a and b are adjacent sides
Square	a², where a = side
Parallelogram	ab sin θ, where a and b are adjacent sides, and θ is the angle between the sides
Trapezoid	½(a + b)h, where a and b are the parallel sides, and h is the altitude
Quadrilateral	½ab sin θ, where a and b are the diagonals, and θ is the angle between them
Regular polygon	¼nl² cot ${180^\circ}\over{n}$ , where n is the number of sides, each of length l
Circle	πr², where r = radius
Ellipse	πab, where a and b are semiaxes
Sphere	4πr², where r = radius
Spherical triangle	(A + B + C − π)r², where A, B, and C are angles (radians), and r is the radius

Business Dictionary: Area

1. Two-dimensional space defined by boundaries, such as floor area, area of a lot, and market area.

2. Scope or extent, as in area of expertise of a professional.

Real Estate Dictionary: Area

A two-dimensional space defined by boundaries; such as floor area, area of a lot, and market area.
Example: A rectangular lot is 50 x 100 feet in dimension. Its area is 5,000 square feet or 0.11 acre.
Example: There are 20,000 square feet of vacant office space in the market area.
Example: A large amount of the town's area is within the floodplain.

Thesaurus: area

noun

A part of the earth's surface: belt, district, locality, neighborhood, quarter, region, tract, zone. Informal neck of the woods. See territory.
A surrounding site: locality, neighborhood, vicinity. See near/far/distance, place.
A rather small part of a geographic unit considered in regard to its inhabitants or distinctive characteristics: district, neighborhood, quarter (often uppercase). See territory.
A sphere of activity, experience, study, or interest: arena, bailiwick, circle, department, domain, field, orbit, province, realm, scene, subject, terrain, territory, world. Slang bag. See territory.

Dental Dictionary: area

Region.

Geography Dictionary: area

One of the basic terms of reference of spatial analysis; the other two are point and line. It may be defined as the extent of a surface or expanse of land, now usually measured in metric square units.

Architecture: area

1. Measurement of surface within specified boundaries.
2. Space either within or outside a structure or location, designated for a specific purpose, as recreation and/or parking area.
3. An uninterrupted interior space.
4. An areaway.
5. The cross-sectional area of steel reinforcement.

Columbia Encyclopedia: area,

measure of the size of a surface region, usually expressed in units that are the square of linear units, e.g., square feet or square meters. In elementary geometry, formulas for the areas of the simple plane figures and the surface areas of simple solids are derived from the linear dimensions of these figures. Examples are given in the table entitled Formulas for Various Areas. The areas of irregular figures, plane or solid, can be computed or closely approximated by the use of integral calculus.

Veterinary Dictionary: area

Pl. areae, areas [L.] a limited space or plane surface.

association a's — areas of the cerebral cortex (excluding primary areas) connected with each other and with the neothalamus; they are responsible for higher mental and emotional processes, including memory, learning, etc.
Brodmann's a's — specific occipital and preoccipital areas of the cerebral cortex, distinguished by differences in the arrangement of their six cellular layers, and identified by numbering each area.
cardiogenic a. — in the embryo includes heart and pericardial rudiments.
central retinal a. — the area of the retina, dorsal to the optic papilla, along the optical axis. Here the retinal vessels are missing.
a. cerebrovasculosa — in anencephaly the cerebral hemispheres are replaced by a sheet of tissue composed largely of blood vessels called the area cerebrovasculosa.
a. cribrosa — that part of the renal crest or renal papilla at which the papillary ducts open into the pelvis.
germinal a., a. germinativa — embryonic disk.
a. medullovasculosa — the central part of a spinal meningomyelocele. It is a raised, reddish protuberance devoid of skin and consists of spinal cord with a surrounding vascular network.
motor a. — that area of the cerebral cortex which, on brief electrical stimulation, shows the lowest threshold and shortest latency for the production of muscle movement.
a. nuda — an area on the surface of a viscus that has no serosal covering.
olfactory a. — 1. the part of the piriform lobe of the brain associated with olfaction.
a. opaca — the opaque area of the embryonic disk of the fertilized avian egg surrounding the area pellucida; it forms some extraembryonic structures.
a. pellucida — the clear central part of the developing embryonic disk in a fertilized avian egg. Produces the embryo's tissues.
a. piriformis temporalis — the cortical area of the piriform lobe of the brain.
primary a. — areas of the cerebral cortex comprising the motor and sensory regions.
psychomotor a. — motor area.
a. sampling — see area sampling.
silent a. — an area of the brain in which pathological conditions may occur without producing clinical signs.
vocal a. — the part of the glottis between the vocal cords.

Word Tutor: area

IN BRIEF: The amount or size of a surface or place.

The area was only about one hundred square feet.

Tutor's tip: A soprano may sing an "aria" (operatic solo) in an outdoor "area" (extent of space) in the summertime.

Wikipedia: area (geometry)

Area is a quantity expressing the size of a figure in the Euclidean plane or on a 2-dimensional surface. Points and lines have zero area, although there are space-filling curves. Depending on the particular definition taken, a figure may have infinite area, for example the entire Euclidean plane. In three dimensions, the analog of area is called a volume.

How to define area

Although area seems to be one of the basic notions in geometry, it is not at all easy to define even in the Euclidean plane. Most textbooks avoid defining an area, relying on self-evidence. To make the concept of area meaningful one has to define it, at the very least, on polygons in the Euclidean plane, and it can be done using the following definition:

The area of a polygon in the Euclidean plane is a positive number such that:

The area of the unit square is equal to one.
Congruent polygons have equal areas.
(additivity) If a polygon is a union of two polygons which do not have common interior points, then its area is the sum of the areas of these polygons.

But before using this definition one has to prove that such an area indeed exists.

In other words, one can also give a formula for the area of an arbitrary triangle, and then define the area of an arbitrary polygon using the idea that the area of a union of polygons (without common interior points) is the sum of the areas of its pieces. But then it is not easy to show that such area does not depend on the way you break the polygon into pieces.

Nowadays, the most standard (correct) way to introduce area is through the more advanced notion of Lebesgue measure, but one should note that in general, if one adopts the axiom of choice then it is possible to prove that there are some shapes whose Lebesgue measure cannot be meaningfully defined. Such 'shapes' (they cannot a fortiori be simply visualised) enter into Tarski's circle-squaring problem (and, moving to three dimensions, in the Banach–Tarski paradox). The sets involved do not arise in practical matters.

In three dimensions, the analog of area is called volume. The n dimensional analog is defined by means of a measure or as a Lebesgue integral; this is sometimes referred to as content.

Formulas

The area between two graphs can be evaluated by calculating the difference between the integrals of the two functions

Areas of 2-dimensional figures

square: $\ s^2$ (where s is the length of one side).
rhombus (includes "kites"): $\frac{1}{2}Dd$ (where D is the length of one diagonal and d is the length of the other diagonal)
circle: $π r 2$ (where r is the radius)
ellipse: $π a b$ (where a and b are the semi-major and semi-minor axes)
any regular polygon: $\frac{Pa}{2}$ (where P = the length of the perimeter, and a is the length of the apothem of the polygon [the distance from the center of the polygon to the center of one side])
a parallelogram: $B h$ (where the base B is any side, and the height h is the distance between the lines that the sides of length B lie on)
a trapezoid: $\frac{(B + b)h}{2}$ (B and b are the lengths of the parallel sides, and h is the distance between the lines on which the parallel sides lie)
a triangle: $\frac{Bh}{2}$ (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: $\sqrt{s(s-a)(s-b)(s-c)}$ (where a, b, c are the sides of the triangle, and $s = \frac{a + b + c}{2}$ 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 (x₁(y₂-y₃)+x₂(y₃-y₁)+x₃(y₁-y₂) 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) Another approach for a coordinate triangle is to use calculus to find the area.
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 ${1 \over 2} \int_0^{2\pi} r^2 \, d\theta$ .
the area enclosed by a parametric curve $\vec u(t) = (x(t), y(t))$ with endpoints $\vec u(t_0) = \vec u(t_1)$ is given by the line integrals

$\oint_{t_0}^{t_1} x \dot y \, dt = - \oint_{t_0}^{t_1} y \dot x \, dt = {1 \over 2} \oint_{t_0}^{t_1} (x \dot y - y \dot x) \, dt$

(see Green's theorem)

or the z-component of

${1 \over 2} \oint_{t_0}^{t_1} \vec u \times \dot{\vec u} \, dt$

Surface area of 3-dimensional figures

cube: $6 s 2$ , where s is the length of any side
rectangular box: $2 (\ell w + \ell h + w h)$ , where l, w, and h are the length, width, and height of the box
sphere: $4π r 2$ , where π is the ratio of circumference to diameter of a circle, 3.14159..., and r is the radius of the sphere
ellipsoid: see the article
cylinder: $2π r (h + r)$ , where r is the radius of the circular base, and h is the height
cone: $\pi r (r + \sqrt{r^2 + h^2})$ , where r is the radius of the circular base, and h is the height. That can also be rewritten as $π r 2 + π r l$ where r is the radius and l is the slant height of the cone. $π r 2$ is the base area while $π r l$ is the lateral surface area of the cone.
prism: 2 * Area of Base + Perimeter of Base * Height

The general formula for the surface area of the graph of a continuously differentiable function $z = f (x, y),$ where $(x,y)\in D\subset\mathbb{R}^2$ and $D$ is a region in the xy-plane with the smooth boundary:

$A=\iint_D\sqrt{\left(\frac{\partial f}{\partial x}\right)^2+\left(\frac{\partial f}{\partial y}\right)^2+1}\; dx dy.$

Even more general formula for the area of the graph of a parametric surface in the vector form $\mathbf{r}=\mathbf{r}(u,v),$ where $\mathbf{r}$ is a continuously differentiable vector function of $(u,v)\in D\subset\mathbb{R}^2$ :

$A=\iint_D \left|\frac{\partial\mathbf{r}}{\partial u}\times\frac{\partial\mathbf{r}}{\partial v}\right|\;du dv.$

This entry is from Wikipedia, the leading user-contributed encyclopedia. It may not have been reviewed by professional editors (see full disclaimer)

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Translations: Area

Dansk (Danish)
n. - område, areal, fladeindhold, distrikt

idioms:

area code områdenummer
built-up area bebygget område

Nederlands (Dutch)
gebied, oppervlak, ruimte, reikwijdte (b.v. van studie), keldergat, strafschopgebied, ,

Français (French)
n. - région, zone, aire, secteur, superficie, (Mil, Pol) territoire, (fig) domaine, champ, (GB) courette en contrebas, coin (salle à manger, etc)

idioms:

area code (GB) code postal, (US, Télécom) indicatif de zone

Deutsch (German)
n. - Gegend, Gebiet, Bereich, Areal

idioms:

area code Vorwahl, Ortsnetzkennzahl

Ελληνική (Greek)
n. - περιοχή, έκταση, επιφάνεια, εμβαδόν, σφαίρα, τομέας

idioms:

area code ταχυδρομικός κωδικός
built-up area πυκνοχτισμένη περιοχή

Italiano (Italian)
regione, campo, zona, area

idioms:

area code prefisso
built-up area zona urbana
text area formato scritto
type area formato scritto

Português (Portuguese)
n. - área (f), superfície (f), território (m), recinto (m)

idioms:

area code código (m) de área
built-up area área construída
disaster area área (f) de desastre
penalty area área (f) de penalidade
text area área (f) de texto
type area área (f) tipo

Русский (Russian)
территория, участок, район, область, площадь

idioms:

area code телефонный код страны, штата и пр.
built-up area место, развивающееся быстрыми темпами
disaster area район бедствия
penalty area штрафная площадка
text area место, отведенное для текста
type area полоса набора; площадь набора

Español (Spanish)
n. - área, zona, región, terreno, campo, distrito, extensión

idioms:

area code prefijo telefónico

Svenska (Swedish)
n. - yta, område

中文（简体） (Chinese (Simplified))
区域, 面积, 范围

idioms:

area code 区域号码
built-up area 建筑物多的地区, 高楼林立的地区

中文（繁體） (Chinese (Traditional))
n. - 區域, 面積, 範圍

idioms:

area code 區域號碼
built-up area 建築物多的地區, 高樓林立的地區

한국어 (Korean)
n. - 지역, 범위, 면적

日本語 (Japanese)
n. - 地域, 地方, 中庭, 面積, 範囲, 分野, 場, 場所, 地下勝手口

idioms:

area code 市外局番
built-up area 市街地
no go area 進入不可地帯
staging area 部隊集結地

العربيه (Arabic)
‏(الاسم) مساحه, منطقه, نطاق, مجال, ساحه البيت‏

עברית (Hebrew)
n. - ‮שטח, איזור, תחום של פעילות או מחקר, חצר ממנה יורדים למרתף‬

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Best of the Web: Area

Some good "area" pages on the web:

American Sign Language
commtechlab.msu.edu

Math
mathworld.wolfram.com

Shopping: Area

Area Rugs	area rugs
area 51	Sunflower Area Rug

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Copyrights:

		Dictionary. The American Heritage® Dictionary of the English Language, Fourth Edition Copyright © 2007, 2000 by Houghton Mifflin Company. Updated in 2007. Published by Houghton Mifflin Company. All rights reserved. Read more
		Sci-Tech Encyclopedia. McGraw-Hill Encyclopedia of Science and Technology. Copyright © 2005 by The McGraw-Hill Companies, Inc. All rights reserved. Read more
		Business Dictionary. Dictionary of Business Terms. Copyright © 2000 by Barron's Educational Series, Inc. All rights reserved. Read more
		Real Estate Dictionary. Dictionary of Real Estate Terms. Copyright © 2004 by Barron's Educational Series, Inc. All rights reserved. Read more
		Thesaurus. Roget's II: The New Thesaurus, Third Edition by the Editors of the American Heritage® Dictionary Copyright © 1995 by Houghton Mifflin Company. Published by Houghton Mifflin Company. All rights reserved. Read more
		Dental Dictionary. Mosby's Dental Dictionary. Copyright © 2004 by Elsevier, Inc. All rights reserved. Read more
		Geography Dictionary. A Dictionary of Geography. Copyright © Susan Mayhew 1992, 1997, 2004. All rights reserved. Read more
		Architecture. McGraw-Hill Dictionary of Architecture and Construction. Copyright © 2003 by McGraw-Hill Companies, Inc. All rights reserved. Read more
		Columbia Encyclopedia. The Columbia Electronic Encyclopedia, Sixth Edition Copyright © 2003, Columbia University Press. Licensed from Columbia University Press. All rights reserved. www.cc.columbia.edu/cu/cup/ Read more
		Veterinary Dictionary. The Veterinary Dictionary. Copyright © 2007 by Elsevier. All rights reserved. Read more
		Word Tutor. Copyright © 2004-present by eSpindle Learning, a 501(c) nonprofit organization. All rights reserved. eSpindle provides personalized spelling and vocabulary tutoring online; free trial. Read more
		Wikipedia. This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Area (geometry)". Read more
		Translations. Copyright © 2007, WizCom Technologies Ltd. All rights reserved. Read more

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area

How to define area

Formulas

Areas of 2-dimensional figures

Surface area of 3-dimensional figures

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