Area Definition and 77 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. M

    My proof of the Geometry-Real Analysis theorem

    Consider a convex shape ##S## of positive area ##A## inside the unit square. Let ##a≤1## be the supremum of all subsets of the unit square that can be obtained as disjoint union of finitely many scaled and translated copies of ##S##. Partition the square into ##n×n## smaller squares (see...
  2. R

    I Vacuum force factors (vacuum created by a "flow through" liquid)

    Greetings all, I'm new here and hope I'm asking this in the correct thread. So, the question is; where you have a vacuum created by a "flow through" liquid witin a large diameter container exerting suction force upon a smaller diameter input tube submerged in a liquid, does the surface area of...
  3. Ahmed1029

    I Triangles on spheres

    If I have a triangle on a sphere with two of its angles 90 degrees each, do I conclude that it's isosceles and that the shortest distance (on the sphere) beteeen the base and the vertix of the thid angle is 1/4 the circumference of a great circle on the sphere? This is the picture I have in...
  4. BlackPhysics

    A cylinder with cross-section area A floats with its long axis vertical

    Summary: A 5.0- cm -diameter cylinder floats in water. How much work must be done to push the cylinder 11 cm deeper into the water? F =Aρgx A 5.0- cm -diameter cylinder floats in water. How much work must be done to push the cylinder 11 cm deeper into the water? F =Aρgx x being the...
  5. V

    I Area of a circle

    i can write the equation of circle easy enough, x^2+(y-r)^2=r^2. i get A=r^2/2 * asin((y-r)/r) + (y-r)/2 * sqrt(r^2 - (y-r)^2) through integration (using change of variable). Letting u = (y-r) and u^2=(y-r)^2, du= dy. Here's the rub... it's not right... :-) Appreciate and thanks in...
  6. P

    Understanding a Velocity-Time Graph

    Summary:: I think we are still in the earlier parts of Physics and I am confused at how "values" work for a velocity-time graph. We are using the formulas to solve an area of a triangle and rectangle to find the total displacement. If a diagonal line begins from above and continue to go down...
  7. person123

    Why are two pieces of wood stronger when bound together?

    The answer learned in class is that the two 2*4s are able to distribute the load over both of them, but I don't think this is an actual answer because that's balanced by the fact that each block is half the area. Does anyone know of the reason for this observation? Thanks!
  8. M

    How do I read this equation for air friction/drag on an object?

    I am trying to understand an excerpt from an article describing the vibrations of a string (eg. guitar/piano) which reads as follows: This is basically the wave equation with Δm representing a small piece of mass from an interval of the string and two forces added to the right side. He...
  9. S

    Using a determinant to find the area of the triangle (deriving the formula)

    This is the question. The following is the solutions I found: I understand that the first line was derived by setting one vertex on origin and taking the transpose of the matrix. However, I cannot understand where the extra row and column came from in the second line. Can anyone explain how...
  10. S

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

  11. T

    I Area of an inclined surface with respect to the original surface

    Hi, I have a problem with inclined planes. The idea is to calculate the stress in an inclined plane of a bar under tension for which you need the surface. I have no idea how this surface is derived, though. In the attached file, you can see what I mean. For a rectangular cross-section, it's...
  12. A

    Neutrinos and their detection

    Assuming that this sphere has a radius of 50kpc, I've converted to m (1.543e21) and plugged into the area equation for a total area of 2.992e43 m^2. From here I've talked myself into circles, and I honestly don't know where to go next. Any help or guidance would be greatly appreciated!
  13. 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...
  14. J

    B Is this true? The area of a circle can be approximated by a polygon

    Hello everyone! I have been looking for a general equation for any regular polygon and I have arrived at this equation: $$\frac{nx^{2}}{4}tan(90-\frac{180}{n})$$ Where x is the side length and n the number of sides. So I thought to myself "if the number of sides is increased as to almost look...
  15. SamRoss

    I Is there an algebraic derivation of the area element in polar coordinates?

    There is a simple geometric derivation of the area element ## r dr d\theta## in polar coordinates such as in the following link: Is there an algebraic derivation as well beginning with Cartesian coordinates and using ##...
  16. Drioton

    How do I find the area of the region bounded by following?

    Using integrals, consider the 7 requirements: Any my attempted solution that I have no idea where I am going: And the other one provides the graph:
  17. S

    Area of a bounded region using integration

    In Calculus II, we're currently learning how to find the area of a bounded region using integration. My professor wants us to solve a problem where we're given a graph of two arbitrary functions, f(x) and g(x) and their intersection points, labeled (a,b) and (c,d) with nothing else given. I...
  18. CrosisBH

    Finding an area of a triangle formed by three points

    Homework Statement P(3, 0, 3), Q(−2, 1, 5), R(6, 2, 7) (a) Find a nonzero vector orthogonal to the plane through the points P, Q, and R. (b) Find the area of the triangle PQR Homework Equations A = \frac{1}{2}|\vec{AB}\times\vec{AC}| Source...
  19. Zack K

    Calculating area using sigma notation

    Homework Statement The question involves using sigma notation of Riemann sums to find the area under the graph of ##x^2+\sqrt {1+2x}##. I managed to calculate most of the values and I have ##16+\frac 8 3 + \Sigma {\frac 2 n \sqrt {9 + \frac {4i} n}}## Homework Equations [/B] ##\Sigma i= \frac...
  20. J

    Minimum possible period

    Homework Statement A particle moves periodically around an ellipse of equation ##\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1##. You can assume ##a>b##. The ##x## and ##y## components of the particle's velocity can never exceed ##v## at any point. What is the minimum possible period of the...
  21. Parallelogram Area Formula Origin - YouTube

    Parallelogram Area Formula Origin - YouTube

    This is my first video!
  22. Jesse Niekamp

    Bending Stress on a Protective Cover

    Homework Statement A 30kg pallet falls from a height of 2m onto a protective cover. The cover has an area of 269m² and an overall thickness of 10.9mm. a. What is the force/pressure on the cover? b. What is the bending stress on the cover? Homework Equations F=ma bending stress =...
  23. YoungPhysicist

    I Area divided by linear function

    The original problem for anyone that can read Chinese: The problem defines a convex polygon with multiple points located in the first quadrant and the required task is to find a linear function y = ax that can spilt the polygon into two parts each...
  24. Bob Walance

    B Our universe's entropy/size vs Bekenstein's theory

    Jacob Bekenstein asserts that the entropy of a black hole is proportional to its area rather than its volume. Wow. After watching Leonard Susskind's video 'The World as a Hologram', it seems to me that he's implying that we are all black hole stuff. Perhaps we (our galaxies and their black...
  25. Mathman2013

    Maximum area of a fenced-in half a circle

    . Homework Statement Let imagine you have gras field formed as a semi circle and you want to fence in that area. The fence is connected to a wall, so you only have to fence in the area formed by the semi-circle. You have to use 60 meters of fence bought at a hardware store. Homework...
  26. C

    Pressure on a Piston

    Homework Statement Suppose there is a tank filled with water and a piston of area S exerts a force F on the water. Suppose I divide the water boundary touching the piston to - N small equal " square " molecules. Then , the force on the upper face of each molecule is F/N . Also, the area of...
  27. CollinsArg

    I Surface area of a revolution, why is this wrong?

    Why is this way of thinking wrong?. can't I assume that when Δx tends to zero is a sufficient approximation of what I want to get? It confuses me with the basic idea of integrating a function to get the area beneath a curve of a function (which isn't also as perfect) . PD: I put Δx tends to...
  28. SSGD

    I Area between two closed curves

    I have been trying to find an answer to this problem for some time. So I was hoping the community might be able to point me in the right direct. I couldn't get the image to load. So above is a link to an image of the problem...
  29. E

    I How to plot a scaled lognormal function

    Hi, I am trying to plot a lognormal function. I have the value of μ=3.5, the value of σ=1.5 and the value of the Area = 1965. I have as well the value of the maximum height (Amp.=4724). I am tryiing to plot these with Excel or with R but I do not know how. I know how to plot a distribution of...
  30. Suyash Singh

    Force due to a chain falling on a table

    Homework Statement a uniform fine chain of length l is suspended with lower end just touching a horizontal table. Find the pressure on the table, when a length x has reached the table.. Homework Equations Pressure = force/area The Attempt at a Solution let mass density, m= mass/l...
  31. M

    How to find the area of a triangular region using Green's Theorem

    Homework Statement You have inherited a tract of land whose boundary is described as follows. ”From the oak tree in front of the house, go 1000 yards NE, then 1200 yards NW, then 800 yards S, and then back to the oak tree. Homework Equations Line integral of Pdx + Qdy = Double integral of...
  32. J

    I How to calculate the area under a curve

    I was doing some integral exercises for getting area under the function. I was doing only more simple stuff, like functions that don't go over the same "x area" multiple times, like a quadratic function. My question is how to calculate area of a loop in an equation x^3+y^3-3axy=0 if let's say...
  33. M

    Flux dependant upon the number of turns? (worked solution on left and plain answer on right; they aren't the same and neither take into account the number of turns, which adds to the confusion) Is the book's answer correct? Doesn't...
  34. Mateus Buarque

    Area of Hexagon - Geometry Challenge

    Determine the area of the painted hexagon, knowing that the area of triangle ABC is 120cm^2 IMG Link: I tried using Heron´s formula, but just ended up with a bunch of terms and one more variable. Sidenote: I guess part of it is figuring out that the side lenghts...
  35. W

    Thermal resistance without area

    I'm (self)studying the physics of heat transfer at the moment. My book gives a relationship between heat transfer rate and thermal resistance as ##\phi=\frac {A \Delta T} {R}##. My book is not in English, so hopefully that is not the cause of this misunderstanding. I double checked that heat...
  36. CynicusRex

    Minimum of x+1/x (Perimeter Square < Perim equal area rects)

    Homework Statement Gelfand - Algebra p.115 problem 264: Prove that a square has the minimum perimeter of all rectangles having the same area. Hint. Use the result of the preceding problem. Homework Equations Preceding problem: Prove that a square has the maximum area of all rectangles having...
  37. O

    Second moment of inertia for a bent rectangle

    Hello. I am currently working with a beam with the following cross-section: It consist of three bended sections with the following parameters, alpha = 90 degrees, Thickness = 4 mm, Radius = 50.59 mm. The top section consist of a small triangle and a rectangle. the triangle have a width = 4 mm...
  38. J

    I Pounds force calculation help

    Someone questioned my calculation for force: 200 PSI Area: panel dimensions are 12" x 18" what is the calculated force in pounds
  39. W

    Friction on a flat rotating surface

    If I push an object such as a cylinder of wood along a flat table (flat face of cylinder in contact with the table) through it's center of mass, the friction or energy required is not dependent of the surface area the block makes with the table, Friction = μ N, correct? And the energy required =...
  40. T

    Integrating with respect to area? Past paper question

    This isn't exactly homework or coursework, it is a past paper question that I cannot find a solution to (my university doesn't like releasing answers for some reason unknown to me). The question is attached as an image (edit: the image displays while editing but not in the post, so I'll try to...
  41. N

    Tension in a Rope

    Homework Statement So it's given the pipe has a inside diameter of 60cm and outside diameter of 70cm. the two ropes AC and AB are separated by a spreader bar. Wants us to find tension in the ropes. Also give is the density of concrete which is 2320kg. Homework Equations pi(r)^2*L=Area The...
  42. caters

    I Area of Appolonian Gasket

    Here is my formula for the area of n layers of appolonian gasket(assuming no circles past the nth layer): $$πR^2 - (πR^2 - (\sum_{0}^{n} x_n*πr_{n}^2))$$ Here R is the radius of the outer circle, r is the radius of an inner circle, x is a function that represents the number of circles in a...
  43. A

    On what area is the pressure of the gas acting?

    Hi I was perplexed as to why the area on which the pressure acts is 'piR^2'. Since one complete half of the sphere is in contact with the gas, hence the pressure should be 4piR^2/2 (half of the surface area of sphere i.e 2piR^2)
  44. steele1

    Prove area of triangle is given by cross products of the vertex vectors....

    Homework Statement The three vectors A, B, and C point from the origin O to the three corners of a triangle. Show that the area of the triangle is given by 1/2|(BxC)+(CxA)+(AxB)|. Homework Equations The Attempt at a Solution I know that the magnitude of the cross product of any two vectors...
  45. N

    I Re-derive the surface area of a sphere

    Hey everyone, I've been stuck on this one piece of HW for days and was hoping someone could help me. It reads: The surface area, A, of a sphere with radius R is given by A=4πR^2 Re-derive this formula and write down the 3 essential steps. This formula is usually derived from a double...
  46. W

    Fluid flow/heat transfer Area dilemma

    Homework Statement I have a confusion regarding areas. Usually in fluid flow I am using the A=(pi*D^2)/4 However in heat transfer we usually use A=pi*D*L Could you please explain this?I mean in first case(fluid flow) it is cross sectional area and in the 2nd case(heat transfer) is like the...
  47. S

    Thermal expansion of liquid in a tube

    Homework Statement A cylindrical glass tube (linear thermal expansion coefficient ##\alpha##) contains liquid (volume thermal expansion coefficient ##\beta##). The height of the tube is ##h_{t,0}## and the height of the liquid inside of it is ##h_{l,0}##. If the temperature changes of an amount...
  48. V

    Resistance Between Two Rectangular Electrodes

    Hello everyone. I have what is probably a relatively simple question. I'm trying to calculate the resistance between two rectangular copper plates submerged in water. I found this thread that briefly discusses it...
  49. I

    Explain why this is correct (Optimization Problem)

    Homework Statement A piece of wire, 100 cm long, needs to be bent to form a rectangle. Determine the dimensions of a rectangle with the maximum area. Homework Equations P = 2(l+w) A = lw The Attempt at a Solution This is what I don't understand, the solutions that I saw from looking around...
  50. V

    Help with force & pressure on spine while bench pressing weights

    Hey everyone. I've never been very good at math/science so I'm seeking a little help from this forum in hopes that there is someone out there who can provide me or guide me to an answer. What I'm trying to figure out is the difference in force or pressure exerted on the human spine for an...