Area Definition and 1000 Threads

Area is the quantity that expresses the extent of a two-dimensional region, 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 length of a curve (a one-dimensional concept) or the volume of a solid (a three-dimensional concept).
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 m2), 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 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 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 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. 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 that such a function exists.

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  1. Z

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  2. M

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  3. A

    I Area of the event horizon of a rotating black hole

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  4. Const@ntine

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  5. A

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  6. Kevin Shen

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  7. Mateus Buarque

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  8. E

    Maximum area for inscribed cylinder

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  9. Delta Force

    Did the Chicxulub Area Contribute to Oil Reserves?

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  10. Saracen Rue

    B Area under the curve of a Polar Graph

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  11. I

    MHB What is the Area of a Quadrilateral with Given Coordinates?

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  12. D

    Area between 2 curves, Volume around X and Y, Centroid

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  13. K

    Area between two graphs as a sum

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  14. dkotschessaa

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  15. W

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  16. K

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  17. J

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  18. M

    MHB How Much Netting Does Rita Need to Cover a Rectangular Area?

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  19. M

    I Area under curve using Excel question

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  20. K

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  21. B

    MHB Why Do I Get Negative Value Integrating y=-50e^-5x?

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  22. Sirsh

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  23. B

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  24. CynicusRex

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  25. Brage Eidsvik

    I Finding Area of Graphs with x^2: A General Solution?

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  26. A

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  27. lfdahl

    MHB What is the total area of the infinite number of inscribed squares?

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  28. core1985

    Help needed in sign of area element -- how do we take sign

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  29. O

    Second moment of inertia for a bent rectangle

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  30. Steven Hansel

    B How do you know the size/maximum area of a gravity field

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  31. I

    Maximum area of triangle inside a square

    Homework Statement ##ABCD## is a square piece of paper with sides of length 1 m. A quarter-circle is drawn from ##B## to ##D## with center ##A##. The piece of paper is folded along ##EF##, with ##E## on ##AB## and ##F## on ##AD##, so that ##A## falls on the quarter-circle. Determine the maximum...
  32. Dileep Ramisetty

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  33. R

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  34. pairofstrings

    B What area of maths is common to all other areas of maths?

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  35. J

    I How much force is exerted at 200 PSI with a panel size of 12 x 18?

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  36. haushofer

    I Area ellipse: parametric form, angles and coincidences

    Dear all, I have a question regarding the computation of the area of an ellipse. The parametric form of the ellipse with axes a and b is $$x(t) = a\cos{(t)}, \ \ \ y(t) = b\sin{(t)} $$ Using this to evaluate the area of the ellipse, usually one takes one halve or one quarter of the ellipse...
  37. Arman777

    What is the Correct Integral for Finding the Area Below y=0 and Above y=lnx?

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  38. W

    Friction on a flat rotating surface

    If I push an object such as a cylinder of wood along a flat table (flat face of cylinder in contact with the table) through it's center of mass, the friction or energy required is not dependent of the surface area the block makes with the table, Friction = μ N, correct? And the energy required =...
  39. Cocoleia

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  40. S

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    Homework Statement Find the area of the region enclosed by the parametric equation x = t3- 7t y = 8t2 Homework Equations A = ∫ y(t) x'(t) dt The Attempt at a Solution I initially began with A = ∫ y(t) x'(t) dt And got to ∫24t4-56t2dt and then to 24∫t4dt - 56∫t2dt but without a limit/defined...
  41. W

    Quantum General textbook within the area of physics

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  42. K

    MHB What Is the Ratio of Area to Perimeter for a Rug with Length 9w?

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  43. S

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  44. W

    Curvature/surface area of a tractrix

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  45. T

    B About the surface area of a prolate ellipsoid

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  46. K

    I Surface formed by moving area along curved axis

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  47. N

    MHB Find value of x and area of parallelogram: a,b,c,P

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  48. J

    I Why is the area of a circular cap on a sphere not equal to πr^2?

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  49. L

    Electrostatic/electromagnetic noise and conductive area

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  50. W

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