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. harryiverson

    First moment of area and second moment of area

    i am perplexed as to the first moment of area and second of area; i would like to know 1. why they come (how they are figured out and distinguished from each other) 2. what is meaning of these 2 moment of area in terms of physics what i have learned is that the first moment of area is used to...
  2. O

    MHB Dimension of the cut-out squares that result in largest possible side area

    A topless square box is made by cutting little squares out of the four corners of a square sheet of metal 12 inches on a side, and then folding up the resulting flaps. What is the largest side area which can be made in this way? What information I have so far is that since the side of the...
  3. andylatham82

    B What does the scalar product of two displacements represent?

    Hi, This feels like such a stupid question, but it's bugging me. Two displacements can be represented with two vectors. Let's say their magnitudes are expressed in metres. The scalar (dot) product of the two vectors results in a value with the units of square metres, which must be an area. Can...
  4. Monoxdifly

    MHB How Do I Find the Area of a Triangle with Non-Intersecting Vertices?

    What's the area of the triangle? It's hard because the vertices aren't in the intersections of horizontal and vertical lines, so I have a hard time determining the side lengths, and it's also for Elementary Students Math Olympiads too.
  5. Frigus

    Why the work done is the area enclosed by the graph of F versus x on x Axis?

    Why work done is area enclosed by graph of F v/s x on x Axis but not y axis. Suppose we apply a force on object which is proportional to displacement as ##\vec f##=## \vec x##²then area enclosed by Force and displacement on x Axis is integral of ##\vec x##²but on y-axis it should be integral of...
  6. karush

    MHB 4.5.1 AP Calculus Exam .... area of piece wise funcion

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  7. L

    B Calculating Triangle Area in Relativity Theory

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  8. jisbon

    Thermal expansion (surface area)

    Hi all, For area expansion, I know the equation goes like: Hence, my answer to part i is ##\begin{aligned}A=A_{0}\left( 1+2\alpha \Delta T\right) \\ = 52\left( 1+2*24\times 10^{-6}\right) \left( 100\right) \\ =52.2496cm^{2}\end{aligned} ## Now I am unsure how to proceed with part 2 in this...
  9. chriscarson

    Finding the the area of the surface at one end of the steel

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  10. Kaushik

    Why does an increase in surface area lead to a reduction in pressure?

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  11. WMDhamnekar

    MHB Maximum area of rectangle which circumscribes the given quadrilateral.

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  12. T

    Quick Method of Calculating the 2nd Moment Of Area Of an I Beam

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  13. karush

    MHB 4.1.310 AP calculus Exam Area under to functions

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  14. T

    Why Use the Parallel Axis Theorem with the Second Moment of AREA?

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  15. T

    Understanding the Value of the Second Moment Of Area

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  16. tflahive

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  17. binbagsss

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  18. S

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  19. H

    Engineering How to find the first area moment of inertia (Q) for an I-beam with flanges

  20. askcr9

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  21. C

    Element of surface area in spherical coordinates

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  22. Z

    I Area Differential in Cartesian and Polar Coordinates

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  23. Saracen Rue

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  24. Wi_N

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  25. karush

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  26. J

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  27. A

    Maximum of the First Moment of Area

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  28. Arman777

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  29. E

    MHB Problem solving calculating area

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  30. Y

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  31. edmund cavendish

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  32. S

    B Anyone here in the CA blackout area that has enjoyed dark skies?

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  33. Mathman2013

    Optimization problem (Max area of a combined semi circle and a square)

    A figure is made from a semi circle and square. With the following dimensions, width = w, and length = l. Find the maximum area when the combined perimiter is 8 meter. I first try to construct the a function for the perimeter. 2*l + w + 22/7*w/2 = 8 - > l = 4 - (9*w)/7 Next I insert this...
  34. grandpa2390

    Find the area of this larger circle

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  35. D

    Engineering Determine a suitable cross sectional area for strength

    Lenght=300mm, Force at the end of the handlebar is 200N What i would like to know is: does that 20x20mm end piece affect the calculation process in any way? and whether there are more than 3 types of stresses in this case. First stress being moment created by the 200N and second stress is shear...
  36. F

    I Why is hysteresis loss proportional to area hysteresis loop ?

    Why does hysteresis loss cause heat and why the heat proportional the area of hysteresis loop?
  37. A

    Earth-like planet with 2x surface area, but same surface gravity?

    How feasible is such a planet? Can the planet still be dense enough to be rocky and not gaseous?
  38. karush

    MHB 4.1.237 AP calculus exam find area

    $\tiny{237}$ $\textsf{The total area of the region bounded by the graph of $f(x)=x\sqrt{1-x^2}$ and the x-axis is}$ $$(A) \dfrac{1}{3}\quad (B) \dfrac{1}{3}\sqrt{2}\quad (C) \dfrac{1}{2}\quad (D) \dfrac{2}{3}\quad (E) 1 $$ find the limits of integration if $$f(x)=0 \textit{...
  39. D

    MCNP6.2: F2 tally and problem with the area calculation

    Hello, after many simplifications my geometry has become very simple: just a box of concrete with a cylinder of steel inside. The source is outside in the air. The cell and the surface cards are like the following: C ******************Block 1: Cells********************** 100 0 99 imp:p=0 99 1...
  40. L

    MHB Year 10 Maths Find the length and width that will maximize the area of rectangle

    The question is in the image. Working out with every step would be much appreciated.
  41. ramadhankd

    Shredder Cutting Area Calculation

    Hello everyone, I'm trying to design a plastic shredder machine, but I'm stuck on how to determine the cutting area of my shredder. I've already made some research, and I think that the cutting area depends on the blade thickness and the plastic thickness. As for why the blade thickness is...
  42. S

    I Calculus- Area between two curves (minimize it)

    Hi, This is my first question here, so please apologise me if something is amiss. I have two curves such that Wa = f(k,Ea,dxa) and Wb = f(k,Eb,dxb). I need to minimize the area between these two curves in terms of Eb in the bounded limit of k=0 and k=pi/dx. So to say, all the variables can...
  43. jim mcnamara

    News Area 51 "Invasion": Over Half a Million Sign Up

    (probably) Delusional people have signed up in droves to go en masse into Area 51. As a joke I hope. https://www.livescience.com/65899-area-51-summer-raid.html https://www.sltrib.com/news/nation-world/2019/07/13/half-million-people/ This place is also called Nevada Test Site. It is the place...
  44. igorrn

    Area between two curves (x = cos(y) and y = cos (x))

    I tried this: X = cos(y) → y = arccos(x) for x E(-1,1) and y E (0,2) Then: There's a point I(Xi,Yi) in which: Cos(Xi) =Arccos(Xi) Then I said area1 (file: A1) A1 = ∫cosx dx definite in 0, Xi And A2 (file:A2): A2 = ∫cosy dy definite in 0, Yi And the overlapping area as A3 (file: A3): A3 = ∫Yi dx...
  45. G

    MHB Smallest possible area of triangle

    Hi, I'm trying to work out this question, and the answer I'm coming up with isn't right. Can anyone help me understand the calculation used to work this out?
  46. Monoxdifly

    MHB [ASK] Find the ratio of the area of triangle BCH and triangle EHD

    A parallelogram ABCD has angle A = angle C = 45°. Circle K with the center C intercept the parallelogram through B and D. AD is extended so that it intercepts the circle at E and BE intercepts CD at H. The ratio of the area of triangle BCH and triangle EHD is ... Here I got that triangle BCH...
  47. J

    Impulse divided by area equals granule strength?

    Summary: I want to crush a single layer of granules from a single material that are 400-600um in size using a flat, circular probe, which will generate a force v time or force v distance curve. The quantity I am trying to measure is the granule (tensile) strength of the material. I believe...
  48. L

    I A random question comes to mind, about the infinitesimal area of rings

    I know the area of a thin ring of radius ##r## can be expressed as ##2\pi rdr##, however, I wonder if I use the usual way of calculating area of a ring, can I reach the same conclusion? I got this: $$4\pi(r+dr)^2-4\pi r^2=4\pi r^2+8\pi rdr+4\pi (dr)^2-4\pi r^2=8\pi rdr+4\pi (dr)^2$$And now I'm...
  49. L

    How to find the area element dA in this situation?

    I know how to find the area of a plane which is parallel to the xy-, yz-, or, xz-plane, those are the easiest case. I also tried to find the area of a plane which is only perpendicular to 1 particular axis plane, like the one passing through points (0,0,a), (a,0,a), (0,2a,0), in which case...
  50. E

    MHB Surface Area of Revolution with Double Integration

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