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. P

    Mean Aerodynamic Chord and Area

    If I have a problem in which the laminar/turbulent transition point is said to be 50% the mean aerodynamic chord, how can I find the area of the wing over which there is laminar flow? Is it simply half the wing area?
  2. anemone

    MHB Find the area of an equilateral triangle

    Show that the curve $x^3+3xy+y^3=1$ has only one set of three distinct points, $P$, $Q$, and $R$ which are the vertices of an equilateral triangle, and find its area.
  3. S

    MHB Understanding the Area of a Curve: A^4 Y^2 = X^4 (A^2 - X^2)

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  4. A

    MHB Stuck trying to integrate the surface area of a curve

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  5. aleksbooker

    Can't integrate the surface area of revolving curve the normal way

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  6. N

    MHB Area under curves and Limit of a sequence,

    Hello, I am looking for an help about this, I have very short time to do many of them and those are an example, could someone show me one solution or explain me how to do it? Thank you if you can help me, I really appreciate. Francesco.
  7. Philosophaie

    Geodesic Equation & Orbital Surface Area Around the Sun

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  8. T

    Integral by interpreting it in terms of area

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  9. A

    Find area under curve and energy of function (matlab)

    Homework Statement For the 1st one you wouldn't really need MATLAB I guess to find the area under the curve, it is 0 and so is its energy. For the 2nd one I got A=1.73 and so E=2.99.Homework Equations area under curve = evaluate integral from t=t1 to t=t2. in this case t=-2 to t=5 since they...
  10. S

    MHB Finding Area Bounded by x^2 & 2x - x^2

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  11. S

    Luminosity and Surface Area

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  12. B

    Finding the Area Bounded by Curves | Quick Solution

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  13. C

    Minimizing area of a semicircle bounded by two lines

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  14. I

    Deriving the surface area equation

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  15. S

    MHB Finding the area bounded by the curves

    So it's been a while since I've done one of these problems. Need to make sure I am using the right procedures to solve it. Q)Find the area bounded by the curve $y = \frac{1}{2}x^2$ and $x^2 + y^2 = 8$ So first thing I did was plug in numbers to get the two graphs. It looks like they intersect...
  16. A

    How do I calculate the number of collisions per unit area in an ideal gas?

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  17. MarkFL

    MHB Area of a Cardioid - Littlehime's Question @ Yahoo Answers

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  18. C

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  19. C

    Surface area from bands: Calculus

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  20. MarkFL

    MHB Minimizing Lateral Area of a Cone with Fixed Volume | Leprofece

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  21. I_am_learning

    Area of a general n-sided polygon

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  22. J

    Fanno flow constant area duct Re number

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  23. D

    Area of a Parametrized Surface

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  24. IridescentRain

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  25. MarkFL

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  26. L

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  27. ShayanJ

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  28. B

    Area element vector for parametric surface integrals

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  29. J

    Is computational physics a PhD area of study?

    If someone wanted to go into computational physics, would that be the PhD area of study, or is it just a branch of the better known areas, like hep, condensed matter, etc. Also is it considered theoretical or experimental? (I would assume experimental). Thanks!
  30. Albert1

    MHB Area of Circle O: Calculate Here

  31. M

    Finding cross-sectional area from terminal velocity.

    Homework Statement The aerodynamic drag on an object moving through air is proportional to Av^2, where A=cross-section, and v=velocity. The terminal velocity of a person without a parachute falling through the air is about 56m/s. Estimate the area of the cross-section of a person seen from the...
  32. MarkFL

    MHB Compute Area of 2 Simultaneous Inequalities: Don Leon at Yahoo Answers

    Here is the question: I have posted a link there to this thread so the OP can view my work.
  33. E

    Focusing on a specific area of Mechanical Engineering Undergrad?

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  34. Saitama

    MHB Calculating the Area Between Two Curves: A Shorter Method?

    Problem: Calculate the area of region defined by the inequalities: $$-1<xy<1$$ $$-1<x^2-y^2<1$$ Attempt: Although I have solved the problem but I am not very satisfied with the method I used. The graph of region is symmetrical in all the four quadrants so I calculated the area in the first...
  35. ShayanJ

    Choosing the research area in physics

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  36. T

    Really Challenging Area Between Curves Problem

    Homework Statement Suppose C:y=f(x) with f a twice-differentiable function such that f''(x)> 0 for each x on the closed interval [0,a] where a is a positive constant. Suppose T is the tangent line to C at a point P= (r,f(r)) on C where r is in the open interval (0,a). Let A be the area of the...
  37. MarkFL

    MHB Bari's question at Yahoo Answers regarding minimizing the surface area of a silo

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  38. A

    Negative area between two curves

    I've been trying to figure out what a negative area means, but I can't. Homework Statement Calculate the area between f(x) = 3^{x} \, , \, g(x)=2x+1 The attempt to a solution The intersections are located in x=0 and x=1. So I do the integral from 0 to 1. \int_{0}^{1} (g(x)-f(x))dx =...
  39. MarkFL

    MHB Find Area of Lemniscate Bounded by Circle: r^2=6sin(2theta) & r=sqrt(3)

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  40. M

    Surface Area of Solid: f(x)=1/x, [1,∞) about x-axis

    Homework Statement f(x) = 1/x Interval [1, ∞) about the x-axis Set-up the integral for the surface area of the solid Then use the substitution u = x2 and integrate using the formula: ∫ sqrt(u2 + a2) / u2 du = ln(u + sqrt(u2 + a2) - sqrt(u2 + a2) / u + C a is a constant Homework...
  41. MarkFL

    MHB Kylie's question at Yahoo Answers regarding minimizing the area of a triangle

    Here is the question: I have posted a link there to this thread so the OP can view my work.
  42. K

    Alternative way to calculate the area of a right angled triangle

    Hello all! New to the forums, and I have a question for you. In my classes, we have been dealing a lot with proofs lately, so when I was working on an assignment, I figured I would try and find my own proof for something, just for the hell of it. I decided to tacle the area of a right angled...
  43. K

    Using sigma sums to estimate the area under a curve

    Homework Statement My apologies in advance for the messiness of the equations; the computers available to us do not correctly process the LaTex code. I am tasked with estimating the area under the curve f(x)=x2+1 on the interval [0,2] using 16 partitions. Online calculators and my...
  44. B

    Help finding direction of area vector on curved surfaces

    Homework Statement I understand how to determine the direction on rectangular surfaces, however when it comes to a cylinder or sphere or, in the case of the problem i am working on now, for a long rod that has been bent around in a circle i.e a a rectangular ring, I don't know what to do.
  45. V

    Max Area of Triangle with Sides (0,1], [1,2], [2,3]

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  46. anemone

    MHB Proving Area Ratio of Equilateral Triangle Divided by Line

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  47. S

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  48. T

    MHB Surface area limited with two functions

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  49. S

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  50. M

    Calculating Shaded Object Area | Integrating Inner Square

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