What is Area: Definition and 1000 Discussions

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. rbmartel

    I Which area should I use to calculate the force on a submerged surface?

    Let's say we have a tank filled with water only half way up. I want to calculate the force being applied by the liquid on one of the walls, that's F = P.A. For the area (A), should I consider the area of the entire wall (H.L), or only the area of the wall that's in contact with the liquid...
  2. anemone

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  3. P

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  4. anemone

    MHB Find area of shaded region

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  5. WMDhamnekar

    MHB Finding the Area of a Figure Given by an Equation

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  6. person123

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  7. K

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  8. AN630078

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  9. P

    I What is the correct way to calculate the area of the sky in square degrees?

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  10. Frigus

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  11. A

    MHB Find Width of Rectangle with 600yd2 and 140yd Fencing

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  12. A

    MHB Good day, Exam Integrals: volume and area

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  13. anemone

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  14. anemone

    MHB Inequality involving area under a curve

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  15. M

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  16. xyz_1965

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  17. xyz_1965

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  18. xyz_1965

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  19. anemone

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  20. U

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  21. S

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  22. anemone

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  23. D

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  24. D

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  25. Adesh

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  26. Saracen Rue

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  27. anemone

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  28. Mahfuz_Saim

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  29. anemone

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  30. D

    I Partial Surface Area of a Tube

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  31. S

    B Why do shapes with the same area have different perimeters?

  32. A

    B Area of a Circle in an Electron's Hydrogen Atom

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  33. H

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  34. M

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  35. karush

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  36. T

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  37. K

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  38. Astrowolf_13

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  39. Saracen Rue

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  40. A

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  41. archaic

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  42. Joe3502

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  43. Robert Friz

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  44. karush

    MHB 4.1.306 AP Calculus Exam Area under Curve

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  45. M

    MHB Calculate numerically the area

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  46. V

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  47. P

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  48. VVS2000

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  49. R

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  50. V

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