Area Definition and 1000 Threads

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.

View More On Wikipedia.org
Recent contents Top users
  1. titasB

    Integrating with Changing Intervals: Finding the Area Between Two Curves

    Homework Statement Find ∫ f(x) dx between [4,8] if, ∫ f(2x) dx between [1,4] = 3 and ∫ f(x) dx between [2,4] = 4 Homework Equations [/B] ∫ f(x) dx between [4,8] , ∫ f(2x) dx between [1,4] = 3 and ∫ f(x) dx between [2,4] = 4 The Attempt at a Solution We are given ∫ f(2x) dx between...
  2. M

    Solve Geometry Q w/ L and c: Find R in Terms of L and c

    Homework Statement In the attached drawing, find R in terms of L and c. Also, at the bottom of the picture I wrote something wrong. I said c, which equals 0.5, is the arc-length of each semi-circle, but I really meant to say each quarter circle. My bad. I'm not given a number for L so that can...
  3. S

    MHB Circumference and Area of A Circle

    hello expert... How to find the circumference of a circle and area of a circle quickly? it's possible using method like ratio or series like pythagoras theory (3,4,5). or another way? somebody could help me out? cheers... susanto3311
  4. I

    Area of Surface: x^2+y^2+z^2=R^2, z>=h, 0<=h<=R

    Homework Statement Calculate the area of the surface ##x^2+y^2+z^2 = R^2 , z \ge h , 0 \le h \le R## Homework Equations ##A(S_D) = \iint_D |\mathbf r'_s \times \mathbf r'_t|dsdt## where ##S_D## is the surface over ##D##. The Attempt at a Solution We write the surface in parametric form using...
  5. M

    MHB Area of graph where have i gone wrong

    I have been asked to finding the area of the graph of the function F(x)=(3x-\pi)\cos\frac{1}{2}x between x=-\pi and x=\pi using integration by parts to integrate the function I get \int 2(3x-\pi)(\sin\frac{1}{2}\pi)+12\cos\frac{x}{2} when I work out the integral for x=\pi and x =-\pi I get...
  6. J

    Why doesn't Gaus's law count for charges outside the area?

    In Gaus's law when the integral is set up, we don't account for the charge outside the closed area. Why is this? How does this law work when the charges outside are not accounted for and only the charges enclosed is in the equation? I need an explanation why Gaus's law still works for...
  7. M

    MHB Find Area of Triangle with Vertices $(0, 0, 0), (1, 1, 1)$ and $(0, -2, 3)$

    Hello! :o We have a triangle with vertices $(0, 0, 0), (1, 1, 1)$ and $(0, -2, 3)$. We want to find the area. How could we find it?? Do we maybe use the fact that the area of the triangle is the half of the area of the parallelogram?? (Wondering) How do we know that it stands?? How can we...
  8. Calpalned

    Area of a triangle using vectors

    ## 1. Homework Statement Let P = (1,1,1), Q = (0, 3, 1) and R = (0, 1, 4). Find the area of triangle PQR Homework Equations ## \frac {|PQ × PR|}{2} ## = area (The crossproduct divided by two) The Attempt at a Solution I lost my answer key, so I want to check if my final answer of ## \frac...
  9. 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...
  10. U

    Proper distance, Area and Volume given a Metric

    Homework Statement [/B] (a) Find the proper distance (b) Find the proper area (c) Find the proper volume (d) Find the four-volume Homework EquationsThe Attempt at a Solution Part (a) Letting ##d\theta = dt = d\phi = 0##: \Delta s = \int_0^R \left( 1-Ar^2 \right) dr = R \left(1 -...
  11. 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...
  12. S

    Surface Area of Foil: Can It Increase?

    If I crumple and unfold a piece of foil, am I increasing it's surface area? Since it's malleable, if I step on it with golf spikes, without completely piercing it, will it gain surface area?
  13. O

    Surface area of a region of a torus

    What is the simplest way to calculate the surface area of a region of a torus? Please see this diagram: https://www.dropbox.com/s/73eics7x43bgiwm/surface-area-torus.png?dl=0 This is a cross section through a torus, the dashed line is the central axis. I am interested in the external surface...
  14. D

    Maximum area of a triangle inscribed in another triangle?

    Homework Statement [/B] Hello! I have this question which I don't quite know how to solve... ABC is an equilateral triangle - the length of its sides equal to (a). DE is parallel to BC 1. What length should DE be to achieve the largest possible area of triangle BDE? 2. What length should DE...
  15. Dennydont

    Total area of contact between tires and ground?

    Homework Statement A car has four tires in contact with the ground each of which is inflated to an absolute pressure of 3.1×105Nm^-2. If a person of mass 82 kg gets into the car by how much will the total area of contact between the tyres and the ground increase assuming that the tire pressure...
  16. Y

    Dependence of resistance on cross section area

    The electrons flow on the outer surface of the resistor, why then does the resistance of a resistor depend on it's cross sectional area and not on it's perimeter?
  17. C

    Surface area contact with catalyst effect on hydrogen redox

    I have also posted this question here: Relationship between surface area of electrode and reaction rate of hydrogen in fuel cells, but I really need an answer before tomorrow morning so I hope you don't mind me posting it here as well! I am looking at the effects of increasing the surface area...
  18. Prof. 27

    Current through Point or Cross Sectional Area

    Homework Statement I'm finding conflicting statements. One says that current is measured as the amount of charge per time that enters through a cross sectional area. Another says that current is measured as the amount of charge per time that passed through a point. Could someone clarify this...
  19. E

    Area of cylinder sliced by sphere

    Hi! Here is my task: Calculate area of cylinder $$x^{2}+y^{x}=ax$$ sliced by sphere $$x^{2}+y^{2}+z^{2}=a^{2}$$. Here is graph: How to do it? If problem was "Calculate area of sphere $$x^{2}+y^{2}+z^{2}=a^{2}$$ sliced by cylinder $$x^{2}+y^{x}=ax$$" I would solve it using double integrals...
  20. E

    Surface area - Double integrals

    Hi! Here is my task: Calculate surface area of sphere $$x^{2}+y^{2}+z^{2}=16$$ between $$z=2$$ and $$z=-2\sqrt{3}$$. Here are 3D graphs of our surfaces: Surface area of interest is P3. It would be P-(P1+P2), where P is surface area of whole sphere. Is it correct? Here is how I calculated...
  21. Calpalned

    Use of integration to find area

    Homework Statement Find the area enclosed by the curve x = t^2 -2t, y = t^0.5 and the y axis Homework Equations Area of a parametric curve = ∫g(t) f'(t) dt, where g(t) = y and f(t) = x The Attempt at a Solution I believe that the limits of integration by be found by setting x and y equal to...
  22. M

    Finding area in polar coordinates

    I've attached the solution to this post. The question is essentially just asking to find the area in one loop for r = cos[3(theta)]. This seems like a fairly simple question (and answer). I've solved and understand the general integration, but I am just a little uncertain on why exactly...
  23. K

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

    Hi all I need to find the relative stiffnes of certain I beams from here http://tsbluebook.steel-sci.org/EN/Browsers/Main.htm Im assuming all I need to do times the Youngs Modulus by the 2nd Moment of Area In my head I am making it more complicated than it should be so I hope this is all it...
  24. A

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

    Just attempted another exam question. Would you mind correcting me if I am wrong? The question is A triangle ABC has sides of length AB= 3.5 m, BC = 5.1 m and AC = 4.2m. a) Calculate the size of the angle B and the size of the angle C, in degrees correct to 1 decimal place, in each case...
  25. M

    Surface Area Vector in Exterior Algebra 3D

    Hello every one . What is the Surface Area vector form in exterior algebra ,I mean by that the Surface Area vector as an exterior form in 3D , just like the volume form .THANKS
  26. M

    Fluid Dynamics - Momentum Equation for Area Change

    Hello :) My question concerns a control volume with a changing area. The momentum equation: p1A1-p2A2 = ṁ(V2-V1) is applied to the control volume. The image shows the equation found when applying the above momentum equation to the control volume: The bit I'm having difficulty with is the part...
  27. K

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

    Homework Statement [/B] The problem is stated in dutch and dutch is my first language. I will try to translate it all as accurately as possible. Imagine the following two functions: f(x)=x^3-4x^2 and g(x)=2x^2. Algebraically calculate the area in between the two graphs. Homework Equations...
  28. avito009

    Why does surface area of an Event Horizon increase?

    Stephen hawking came to know of a study that stated that surface area of an event horizon increases. So he said that since area has increased Entropy also had increased. But why does surface area of an Event Horizon increase?
  29. anemone

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

    As shown in the figure below, triangle $PQR$ is divided into six smaller triangles by lines drawn from the vertices through a common interior point. The areas of four of these triangles area as indicated. Find the area of triangle $PQR$.
  30. H

    [Double integral] Area of a triangle

    Hi! I'm stuck with the following problem: ----------------------------------- Calculate ∫∫ (x-y)*|ln(x+2y)| dxdy where D is the triangle with corners in the coordinates (0,0), (1,1) and (-3,3) ----------------------------------- I get the following lines that forms the triangle: y=-x, y=x...
  31. O

    Integrating pressure over area to get friction force

    I'm doing some experiments where I need to calculate the resistance force on a cylindrical body (cable) when it's being pulled through water saturated sand We derived formula from a theory which was originally based on a square body by using stress components. This way we know the pressure at...
  32. M

    Area of circle in polar coordinates

    Homework Statement r=2cos(theta) I want to find the area using polar integration. Homework Equations area=(1/2)r^2 from 0-pi The Attempt at a Solution When I plug everything in I get 2pi as the answer. I'm in multivariable calculus so this is very frustrating. What am I doing wrong, I don't...
  33. Just_some_guy

    EMF induced via change in Area

    I have been studying electromagnetism this year and we have spoken about Faradays law of electromagnetic induction and eventually how the emf induced is equal to the negative time rate of change of magnetic flux I noticed however that all examples include a time varying magnetic field, which...
  34. SalfordPhysics

    Comp Sci Fortran90: DO loop for sequence of numbers

    Homework Statement A program finds the area under the Gaussian Distribution Curve between ±σ using Simpsons Rule. Modify the program to investigate the effect of the number of strips. Do this by using a DO loop in the main program for the following sequence of number of strips (n); n-2, n-4...
  35. G

    What is this area of study called?

    Hello all, I am a freshman enrolled in Mechanical engineering. Growing up, I have always been fascinated by the LEGO technic toys and the real-life machines and wondered how do engineers design these machines and mechanisms. When someone design a complex machine or a mechanism, how do they...
  36. L

    Second Moment of Area calculation?

    Hi there, I am covering a topic on welding at university and I am struggling with deriving the second moment area for the structure attached. The desired second moment of area is also given in the attached image. The first part is particularly confusing to me as I know that the total second...
  37. C

    Volume of Solid w/ 4x4 Square Base & Cross-Sections of Semicircles

    Homework Statement Find the volume of the solid whose base is a 4 by 4 square. Cross sections perpendicular to one diagonal of the square base are semi-circles with diameter on the base. Homework Equations V=pi r^2 A=S^2 The Attempt at a Solution The cross sections are perpendicular to the x...
  38. J

    Magnetic flux linkage and area as a vector

    Homework Statement A circular coil of diameter 24 mm has 40 turns. The coil is placed in a uniform magnetic field of flux density 85 mT with its plane perpendicular to the field lines. a) i Calculate the area of the coil ii The flux linkage through the coil. b) The coil was reversed in a...
  39. T

    MHB Finding area between 2 functions

    1. I have to find the area between $x = 2y^2$ and $x = 1 - y$ I find the intersection points $ 1 -y = 2y^2$ $2y^2 + y - 1= 0 $ $(2y - 1)(y + 1)= 0$ so y = 1 and -1 However, x = y - 1 is not a vertical line so I am not sure how 1 and -1 can be intersections. Also, when I plug these...
  40. 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...
  41. 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...
  42. 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...
  43. P

    Can there be an area with no gravity?

    Can there be an area without gravity? Are branes the only places with gravity?
  44. 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...
  45. 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...
  46. 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...
  47. marcus

    ΛEPRL quantiz'n of cosmological horizon area in Planck units

    The ΛEPRL spin foam model presented 25 November at ILQGS by Haggard and Riello achieves an interesting quantization of the cosmological constant. Basically this is done on slide #10 around minute 15 of the audio. http://relativity.phys.lsu.edu/ilqgs/haggardriello112514.pdf...
  48. 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...
  49. 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...
  50. H

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

    This one is kicking my butt. In a circle, find the exact area of a segment bounded by chord \overline{CD} and \stackrel{\frown}{CD}. \overline{CD}=12\sqrt{3} and the \measuredangle \stackrel{\frown}{CD}=120^{\circ}.
Back
Top