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. I

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

    find the area of the region that lies inside the first curve and outside the second curve. $r=3cos(\theta)$, $r=1+cos(\theta)$ $3cos(\theta)=1+cos(\theta)$ $2cos(\theta)=1$ $cos(\theta)=\frac{1}{2}$ $\theta= \frac{\pi}{3}, \frac{5 \pi}{3}$$A=\frac{1}{2} \int_{\pi/3}^{5\pi/3} \ (3cos(\theta))^2...
  2. I

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

    find the exact area of the surface obtained by rotating the given curve about the x-axis. $x=cos^3(\theta)$, $y=sin^3(\theta)$, $0 \le \theta \le \pi/2$. $\frac{dx}{d \theta}=-3sin(\theta)cos^2(\theta)$ $\frac{dy}{d \theta}= 3sin^2 (\theta)cos( \theta)$ $S=2 \pi \int_{0}^{\pi/2} \...
  3. Lebombo

    Area of a triangle (cross product lesson)

    Homework Statement Youtube: Lec 2 | MIT 18.02 Multivariable Calculus, Fall 2007 (Video time frame: between 11:00 minutes and 12:30 minutes) Find the area of a triangle. Area = \frac{1}{2}(base)(height) = \frac{1}{2}|a||b|sinθ The lecturer says to first find cosine of the angle using dot...
  4. J

    Show rectangular box of given volume has minimum surface area when

    Homework Statement show rectangular box of given volume has minimum surface area when the box is a cube [gotta show it with partial derivatives to minimize] Homework Equations surface area = 2(wl+hl+hw) volume = whl The Attempt at a Solution so this is the one I would be minimizing...
  5. A

    MHB Quick Surface Area and Application Question

    For the first one, I'm completely dumbfounded. I used the SA formula for and ended up getting 2 $π\int_{π/6}^{0} \, cos2x * √1+4sin(2x)^2 ) $ And took u = sin2x, ultimately giving me $ π\int_{√3/2}^{0} \,d (1+ 4u^2)^(1/2) $ I'm lost as to what to do next. I'm almost definite that my...
  6. G

    Finding area given definite integral

    Question : https://www.physicsforums.com/attachments/71328 My question is how did the 2a and 2b come from?? Equations: Area of trapezoid =(a+b/2)(h) Attempt: I know that the area of a trapezoid is (a+b/2)(h) However why is there now a 2a and 2b in its place? Could it be related to the 2s...
  7. maverick280857

    How is the Nambu Goto action proportional to the world sheet area?

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  8. Y

    What is the Area of a Transformed Region?

    Homework Statement Let T:R2→R2 be the linear transformation such that T(x,y)→(9x+3y,5x+2y). Let R be the region on the plane defined as {(x,y) \in R2/ 0≤x≤2 and 0≤y≤2 }. Consider the region T(R) \subset R2, which is the image of the region R by the linear transform T. What is the area of the...
  9. S

    Surface area of between 2 cones

    find the area of the cylinder x^2+z^2=a^2 that is inside the cylinder x^2+y^2=a^2. my attempt: parameterise x^2+z^2=a^2 as a vector r(x,y) = (x,y,(a^2-x^2)^1/2). using the formula given here : http://tutorial.math.lamar.edu/Classes/CalcIII/SurfaceArea.aspx, I found the surface area = the...
  10. J

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

    I am working on a task right now. I am currently trying to find the area of a parallelogram. I do not have the height. I only have the dimensions. I have tried suggestions like dividing the parallelogram into triangles and doing 1/2bh. The dimensions I have are the 1/2,1/2,\sqrt{2}/4...
  11. I

    MHB Finding Surface Area from Equation in Terms of x=g(y)

    if I am finding the surface area, and I am given the equation in term of x=g(y) about the x-axis, do i have to solve for y or can i just integrate in terms of y? would i just use dx/dy instead?
  12. P

    Rotational frictions relationship to surface area

    Hi, I have a question I'd like some help with which is related to my job working with drill rigs. Where you have a cylinder such as a drill string where the forces of the ground pressing in from all sides how much influence does the surface area have on the rotational force required to...
  13. Nathanael

    Surface area of the Sphere

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  14. M

    Finding area by using a summation

    Hello everyone, I've been working on an area summation problem in my book for quite a bit and I can't solve it. Find the area under the straight line y=2x between x = 1 and x = 5 The book shows the answer as 24 and Maple does as well, but I'm not getting 24, I'm getting 8. Area summation formula...
  15. I

    Is the Planck Length a 3d area?

    If so, is 'its half' also not a real size of space? What I am asking is pretty much; what is the meaning of the notion of a 'smallest possible' 3d area? What is the meaning of that cusp between 2d and 3d? Almost like asking what is in between 1.999999999999(repeating) and 2. Must...
  16. K

    How can area be represented as a vector in three dimensions?

    How is Area a vector? How does it have direction? I thought it was basically a scalar quantity because it only had magnitudes, e.g. 4m^2, 7m^2 etc. Please help. I can't understand. :(
  17. M

    MHB Area of 'that' part of the circle ....

    Using integral find the area of that part of the circle x^2 + y^2 = 16 which is exterior to the parabola y^2 = 6x.
  18. D

    Finding the area beneath a curve help

    1. Hi, I have been asked to calculate the area beneath a curve. The curve is y = 2x^2 + 7x + 24 and the values of x to find the area within are 5 and 2. 3. My attempt stands as follows: A = 5,2∫ (2x^2+7x+24)dx [ 2x^3/2 + 7x^2/2 + 24x...
  19. K

    A spherical raindrops evaporates at rate proportional to surface area?

    i want to find V(t) At first i found this problem was very simple but when i try to write differential equations i ended up with these V' = kA that's for sure then i confined the problem only to spherical shape and no other shapes of raindrops involved as i can't express A in term of V alone(...
  20. B

    Area of a cylinder inside a sphere (surface integral)

    Homework Statement Find the area of the cylinder x^2 + y^2 -y = 0 inside the sphere x^2 + y^2 +z^2 =1 Homework Equations dA = sec \gamma dydz where sec \gamma = \frac{|\nabla \phi|}{|\partial \phi/ \partial x|} The Attempt at a Solution The method shown in this section is to...
  21. M

    Friction torque vs. contact area

    I promise this isn't homework, it's actually for my research :-P Here goes: When two bodies are in contact and *translate* with respect to each other, there is no dependence of friction on contact area, and Ff = mu*Fn. Now, assume we have a block of wood on a table with mass m and we apply...
  22. A

    Coordinate geometry with area of triangle

    Let A(1, 2), B (3,4), C( x, y) be points such that (x- 1) (x-3) +(y-2) (y-4)=0. Area of triangle ABC=1. maximum number of positions of C in the xy plane is (a) 2 (b) 4 (c) 8 (d) None of these I have tried using the staircase formula which gives me something like x-y=2. Therefore I see only...
  23. J

    Area and volume integral of vector field

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  24. G

    Finding the Area of a Similar Right Triangle

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  25. U

    PhD MUST be on same area as masters degree?

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  26. K

    Integral, why antiderivative is area under curve

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  27. maistral

    Sides of a triangle given an area and an equation.

    WARNING: THIS IS NOT HOMEWORK~! Okay, so the problem goes like this: "Find a,b,c of a triangle; If a+b+c = 10 ; Area = 10" I know it sounds totally vague (I think so too). So I tried using the Pythagorean theorem; c2 = a2+b2 then the given equation; 10 - a - b = c; then the...
  28. adjacent

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

    Homework Statement Find the area of triangle formed by the points ##A(5,2)## , ##B(4,7)## , ##C(7,-4)## Homework Equations Nah The Attempt at a Solution Is there any better way than finding the angle between lines and their lengths and then the area?
  29. A

    Area of a polygon- using numerical integration

    Hi, I need to calculate area of an irregular polygon which can be of any complex shape numerically i.e. using numerical integration techniques. Please can anyone suggest any reference material / best way of going about this efficiently? Akash
  30. F

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

    I learned in my calc 1 class that to calculate the arc length of a curve, we are to compute the integral of the function. For example, the integral of a function that describes the path of a thrown baseball would give the total distance traveled by the baseball (I hope I'm using the term arc...
  31. G

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

    This question has been asked probably many times, and is inspired by area that appears in denominator of energy flux unit [W/m^2]. From what i have read so far, i come to conclusion, that this post should, first of all, try to explain that this question makes some sense. First thing that...
  32. Q

    Area of a Square: Derivation & Integration

    So I was reviewing my random process notes. In it there is an integral that they have that I can't seem to get the right derivation of when they try to simply the math for ergodic mean. Basically, you have the following: A square from (-T,T) on both the x-axis and y-axis. What they want to...
  33. B

    Surface integrals to derive area of sphere

    Given a sphere x^2 + y^2 + z^2 = a^2 how would I derive the surface area by using surface integrals? The method I've tried is as follows: dA = sec\ \gamma \ dxdy where gamma is the angle between the tangent plane at dA and the xy plane. sec \gamma = \frac{|\nabla \varphi|}{\partial \varphi...
  34. S

    Area of a Triangle Using Vectors.

    Homework Statement A triangle has verticies A(-2,1,3), B(7,8,-4), and C(5,0,2). Determine the area of the triange ABC. The correct answer is 35.9 square units. Homework Equations Has to be done by using dot product and/or cross product. Dot product: a(dot) b= |a||b|cos(theta) Cross...
  35. V

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

    Homework Statement The solar constant, the amount of solar energy reaching the Earth per unit time and area above the atmosphere and an element of area perpendicular to the direction from sunlight is 1.36 kW / m². For an element of area whose normal makes a angle q with the direction of...
  36. Albert1

    MHB What is the Area of Triangle ODG?

  37. B

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

    Hello . I am in the end of my exams and i have to do a geometry figure like a pyramid ( view image ) below Now i should find the Perimeter, Volume and Surface of this figure . Lengths are all 5 cm, Can somebody find and write the Permiter,volume and surface for this figure please it's urgent...
  38. A

    Calculating the Area Under an Inexisting Curve?

    Hi, I always wondered, what is the area under an inexisting curve. That arctan(1/sqroot(x^2-1)) for example. Its domain does not include from -1 to 1. If I take its integral from 0 to 10, what answer should I get?
  39. adjacent

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

    Homework Statement We know the if two 2D shapes are similar, the ratio of their areas are equal to the ration of the square of the corresponding sides. ##\frac{A_2}{A_1}=(\frac{l_2}{l_1})^2## Prove this The Attempt at a Solution For rectangles: Let the area of first rectangle be ##A_1## and...
  40. M

    Why does the horizon area of a black hole never decrease?

    Hawking said that the horizon area of a black hole never decreases and illustrated that in his Hawking Are Theorem: dA/dt ≥ 0 Does anyone know why is it like that. Why doesn't the area decrease?
  41. J

    Why is preferable the line integral than the area integral over C plan

    Why is preferable to use the line integral than the area integral over the complex plane?
  42. F

    A Question regarding stress/cross sectional area

    Homework Statement A cylindrical rod has a spherical bubble in it. as illustrated in figure 4.2 (in the attachments) The rod has a cross sectional area of 3.2 x 10^-6 m^2 and is stretched by forces of magnitude 1.9 x 10^3 N. The maximum Stress that the cylinder can take is 9.5 x 10^8...
  43. DocZaius

    Pressure flow as a function of area

    Homework Statement Please note that I am making this problem up and it is not "assigned" to me: A large tank of water has two small holes at the same height, near the bottom. Hole A's area is twice hole B's area. Is the volumetric flow rate (volume per time) of water out of hole A bigger than...
  44. J

    Finding the area between Polar Curves

    Homework Statement Find the area of the region that consists of all points that lie within the circle r = 1 but outside the polar equation r = cos(2θ) Homework Equations A = ∫ 1/2 (r2^2 - r1^2) dθ, where r2 is outer curve and r1 is inner curve. The Attempt at a Solution Here is...
  45. _N3WTON_

    Area Polar Graph Homework: Find Region's Area

    Homework Statement Find the area of the region. Interior of: r = 2 - sin(b) Homework Equations A = 1/2 ∫ (r)^2 dr The Attempt at a Solution I really don't have any idea how to approach this problem. I don't understand how to determine my limits of integration. The only part of the problem I...
  46. J

    Surface area and volume uniquely determine a shape

    Is this so? I cannot think of a counter-example and it is too general a statement to prove.
  47. veronica1999

    MHB The two values for $\theta$ would be 0 and $\pi$.

    Question: What is the area enclosed by the lemniscate r2 = -25cos(Ѳ)? The answer is 25. I can't seem to set up the integral for this question correctly. I know that the area enclosed by a polar curve is obtained by the integral(r2/2) dѲ, but I can't determine what the interval of integration...
  48. M

    Area of a right triangle with little data

    Homework Statement Homework Equations x^2 + y^2 = 9 A = 0.5xy x ≠ y The Attempt at a Solution x^2 + y^2 = 9 A = xy/2 (x + y)^2 = x^2 + 2xy + y^2 = 9 + 2xy = 9 + 4A A = ((x+y)^2 - 9)/4 Then I am lost. I need to find the area.
  49. D

    MHB Double Integral Problem/ Surface Area of parametric surface

    Hi I'm doing a surface area problem with a parametric surface and I got the cross product but I can't figure out the double integral. I found the solution online but with no explanation, so can someone explain how to solve this integral: thank you!
  50. Saitama

    MHB Maximum area of triangle and quadrilateral - given perimeter

    Problem: Let $0<a<b$ i)Show that amongst the triangles with base $a$ and perimeter $a+b$, the maximum area is obtained when the other two sides have equal length $b/2$. ii)Using the result (i) or otherwise show that amongst the quadrilateral of given perimeter, the square has maximum area...
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