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. \Theta

    Scientific Computing Application Area?

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  2. nomadreid

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  3. I

    What is the Formula for Finding the Area of a Circular Coil?

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  4. Dethrone

    MHB Calculating the Area of Region Bound by a Curve and its Asymptotes

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  5. P

    Exploring the Concept of Tangent on a Large Circle

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  6. A

    MHB Calculating Area from Differentials

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  7. Greg Bernhardt

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  8. I

    MHB What is the area of the region between two parametric curves?

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  9. I

    MHB Find Area of Rotated Curve: $x=cos^3(\theta)$, $y=sin^3(\theta)$

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  10. Lebombo

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  11. J

    Show rectangular box of given volume has minimum surface area when

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  12. A

    MHB Quick Surface Area and Application Question

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  13. G

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  14. maverick280857

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  15. Y

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  16. S

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  17. J

    MHB What is the area of a parallelogram without knowing the height?

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  18. I

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  19. P

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  20. Nathanael

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  21. M

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  22. I

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  23. K

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  24. M

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  25. D

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  26. K

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  27. B

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  28. M

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  29. A

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  30. J

    Area and volume integral of vector field

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  31. G

    Finding the Area of a Similar Right Triangle

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  32. U

    PhD MUST be on same area as masters degree?

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  33. K

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  34. maistral

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  35. adjacent

    Area of Triangle: Find Points A, B, & C

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  36. A

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  37. F

    Is the Area Under a Curve Equal to its Arc Length?

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  38. G

    How cross-section area of single E-M wave looks like?

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  39. Q

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  40. B

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  41. S

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  42. V

    What is the solar constant in certain area of earth per day?

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  43. Albert1

    MHB What is the Area of Triangle ODG?

  44. B

    MHB Help with Geometry: Volume, Area and Perimeter of Pyramid

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  45. A

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  46. adjacent

    Is Galileo's square-cube law a universal proof of similarity in 2D shapes?

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  47. M

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  48. J

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  49. F

    A Question regarding stress/cross sectional area

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  50. DocZaius

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