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. T

    MHB What is the area between two intersecting parabolas?

    Find the area of the region enclosed by the parabolas $$y = x^2 $$and $$y = 2x - x^2.$$ So they intersect at 0 and 1. The derivative of $$y = 2x - x^2.$$ is $$\d{y}{x} = 2 - 2x$$ When I plug in 1 I get 0, and when I plug in 0, I get 2, so I subtract 2 from 0 and the area is -2. But the area...
  2. U

    Proper distance, Area and Volume given a Metric

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  3. C

    Why is Normal Area to Light Rays Invariant?

    It is a fact that all inertial observers would measure the same area normal to a beam of light rays in relativity. You can prove this by considering the displacement vector connecting a light ray to its neighbouring light rays. But I wondered if there were some intuitive explanation of why this...
  4. S

    Surface Area of Foil: Can It Increase?

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  5. O

    Surface area of a region of a torus

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  6. D

    Maximum area of a triangle inscribed in another triangle?

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  7. Dennydont

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  8. Y

    Dependence of resistance on cross section area

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  9. C

    Surface area contact with catalyst effect on hydrogen redox

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  10. Prof. 27

    Current through Point or Cross Sectional Area

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  11. E

    Area of cylinder sliced by sphere

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  12. E

    Surface area - Double integrals

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  13. Calpalned

    Use of integration to find area

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  14. M

    Finding area in polar coordinates

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  15. K

    What does Young's Modulus x 2nd Moment of Area Equal

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  16. A

    MHB Given 3 sides of a triangle, compute interior angles and area

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  17. M

    Surface Area Vector in Exterior Algebra 3D

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  18. M

    Fluid Dynamics - Momentum Equation for Area Change

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  19. K

    Area in between graphs, one graph partially below y=0

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  20. avito009

    Why does surface area of an Event Horizon increase?

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  21. anemone

    MHB What is the area of triangle PQR divided into six smaller triangles?

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  22. H

    [Double integral] Area of a triangle

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  23. O

    Integrating pressure over area to get friction force

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  24. M

    Area of circle in polar coordinates

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  25. Just_some_guy

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  26. SalfordPhysics

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  27. G

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  28. L

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  29. C

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  30. J

    Magnetic flux linkage and area as a vector

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  31. T

    MHB Finding area between 2 functions

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  32. T

    MHB Finding Area between 3 functions

    I need to find the area bounded by: $y = \sqrt{x}$, $y = x/2$, and $x = 9$. I found that the intersecting point is 4 and $y = \sqrt{x}$ is the smaller function between 4 and 9 so: $$\int_{4}^{9}\frac{x}{2} - \sqrt{x} \,dx$$ and I get $$ \left[ \frac{x^2}{4} - \frac{2x^{3/2}}{3}\right]_4^9...
  33. T

    MHB Finding Area between 2 functions

    Hi, I have this problem to find the area between 2 curves: $y = x^2$ and $y = \frac{2}{x^2 +1}$ I found that the points of intersection are -1 and 1 and it is symmetrical. I get $2\int_{0}^{1} \ \frac{1}{x^2 + 1} - x^2 dx$which I am unable to solve. I have tried u-substitution but I end up...
  34. K

    Calculate area of subsection of a three-dimensional surface

    For the following three-dimensional surface, z = -4.53 + 2.67x + 2.78y - 1.09xy, I would like to calculate the area for each of three subsections of this surface: (1) for which z is in between the corresponding x and y values (i.e., x < z < y OR y < z < x); (2) for which x is in between the...
  35. P

    Can there be an area with no gravity?

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  36. T

    MHB What is the area between the functions $y = |2x|$ and $y = x^2 - 3$?

    Hi, I need to find the area between these 2 functions: $$y = |2x|$$ and $$y = x^2 - 3$$ So I need to find the points of intersection: $$|2x| - x^2 + 3 = 0$$ for which I get x = 3, -1 However, since there are no negative x values in y = |2x| I get $x = 3, 1$ I find that $y = |2x| $is...
  37. T

    MHB So, the area between the two functions is 72 units squared.

    I need to find the are between $$y1 = 12 - x^2$$ and $$ y2 = x^2 - 6$$. Since y1 is greater, I subtract y2 from y1 getting: $$ \int 18 - 2x^2$$ which is $$18x - 2x^3 / 3$$, The intersecting points are $$x = -3 and x= 3$$. So I find $$18x - 2x^3 / 3 from x = 3 to x = -3$$(I'm trying to...
  38. T

    Find the area of the bounded region

    Hi guys I am very new here this is my second post. (sorry in advance i don't know how to use the functions of the site fully yet) i think this is the correct method to follow, some feedback or hints would be great thanks in advance! 1. Homework Statement Find the area bounded by where...
  39. marcus

    ΛEPRL quantiz'n of cosmological horizon area in Planck units

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  40. B

    Why is radiance defined per projected area normal to the beam direction?

    Radiance is defined as radiant flux per solid angle per projected area normal to the beam direction: ##L = \frac{d^2 \Phi}{d \vec\omega \cdot d A_\perp}## where ##A_\perp = A \cos \theta## and ##\theta## is the angle between the beam direction ##\vec\omega## and the surface normal. I kind of...
  41. M

    Calculate Circle Radius/Diameter from Surface Area

    Hi How is circular surface area values square rooted ??. I am using this formula r2Xπ=mm2 to calculate the surface area of a circle but i want to know if there is a formula to get back to the diameter or the radius of the circle ?. example i have a circular surface area of 5.26mm2 and i want...
  42. H

    MHB What is the area of a segment bounded by a chord and an arc in a circle?

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  43. K

    Relationship between surface area and damping coefficient

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  44. C

    Conceptual problem regarding pressure and surface area

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  45. JasonHathaway

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  46. P

    Find the dimensions of the rectangle of greatest area

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  47. gracy

    Understanding Surface Area: Foldings, Invaginations, Roughness & Powder

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  48. S

    Proof of disk moment of inertia using area density

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  49. L

    Commuting via car or BART (Bay Area Rapid Transit)?

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  50. I

    C/C++ Compute the area of a triangle c++

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