What is Areas: Definition and 246 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.

View More On Wikipedia.org
  1. B

    A Learning about other experimental areas of physics

    Hello! My research is mainly in atomic and molecular physics but from time to time I come across nice experimental setups, that show promise for my work, from other fields, like chemistry or even high energy physics. I was wondering if there is a more organized way to explore experimental setups...
  2. Saracen Rue

    B A Simpler Way to Find the Shaded Area?

    Consider the following scenario: Given that points ##M## and ##N## are the midpoints of their respective line segments, what would be the fastest way to determine what percentage of the squares total area is shaded purple? I managed to determine that the purple shaded area is ##5\text{%}##...
  3. N

    MHB Calculating Sand Volume for a Cylindrical Containment Vessel

    A spherical pressure vessel with a diamter of 10 m is tightly enclosed in a cylindrical containment vessel with the sphere just touching on all 4 sides . Additional protective material (assume is sand ) is added to the cyclinder to provide additional support .How much sand is required so that...
  4. D

    Julia is most mature in what areas?

    Hello, up late toying with Julia programming language. Just curious if anyone here has knowledge on something. In which areas has the Julia programming language gained the most maturity in terms of ecosystem, libraries and practical industry adoption?
  5. karush

    MHB -7.64 Determine the following standard normal (z) curve areas:

    Determine the following standard normal (z) curve areas: Determine the following standard normal (z) curve areas: a. The area under the z curve to the left of $1.75$ from table $5\ \textit{$z^{*}$} =1.7 \textit{ col } .05 = .9599$ $\textit{ \textbf{$W\vert A$} input }...
  6. anemone

    MHB Solving a Primary Math Problem: Find the Areas of A & C

    \begin{tikzpicture} \draw (0,-3) rectangle (6,3); \begin{scope} \draw (6,-3) arc (-90:-270:3cm); \end{scope} \begin{scope} \draw (6,-3) arc (0:90:6cm); \end{scope} \coordinate[label=above: A] (A) at (2,0); \coordinate[label=above: B] (B) at (3.6,2.4); \coordinate[label=above: C] (C) at...
  7. T

    Other Finding academic expertise in specific areas

    I wonder how to find theoretical physics labs which have high level expertise in specific areas. Is there an online DB or "ranking" by area rather than by country? Also looking for HCR in TP. Cheers!
  8. D

    I Double integrals - do areas cancel?

    If i do a double integral of 1.dxdy to find an area of an odd function eg. y=x from +a to -a i get zero because the area below the x-axis cancels with the area above the x-axis. If i do a double integral of a circle centred at the origin i get the area to be πr2 ; so why doesn't the area below...
  9. Astronuc

    Beware of Vibrio bacteria in warm water: Gulf and Atlantic Coast areas

    Bailey's death came days after he handled crabs, one which pinched his finger causing a cut. Some vibrio bacteria entered the fresh wound into the blood stream. Since 2007, reports of illness from toxic forms of vibrio have tripled in South Carolina and nearly doubled in North Carolina...
  10. R

    B Gamma rays over very large areas

    According to this link https://apps.dtic.mil/dtic/tr/fulltext/u2/a351472.pdf, HERMES III at Sandia National Labs can generate very large area gamma rays by converting the electron beam into bremsstrahlung radiations after hitting the tantalum target. As we can see in the isodose contours in the...
  11. karush

    MHB S8.4.2.48 find int given 2 areas

    screenshot to avoid typosok I assume f(x) is the eq of a sloped line.. well at the simplest option at $2\ge x \ge 8$ so $y=\dfrac{7.3}{6}$ but I don't know how to get the slope so $\displaystyle\int_2^4 f(x) = 5.9$
  12. person123

    I Maximum Strain For Samples of Different Cross-Sectional Areas

    I would assume that because the samples are made of the same material they would fail at the same stress and so the same strain. However, the data shows that the sample with a greater cross-sectional area fails at a greater strain, and the two are roughly proportional. Does anyone know what...
  13. E

    Rate of conduction heat transfer with different hot-side and cold-side areas

    When we measure 'the rate of conduction heat transfer'=Q , we assume that the hot side and the cold side's area are same. But if the both side's area is different to each other, how can i know the rate of conduction heat transfer? like below figure. Would you like to help me?? Thanks.
  14. Ygggdrasil

    Celebrating RNA Day: Exploring Exciting Areas of RNA Research

    Today (AUG 1) is RNA Day, an international celebration of one of the most versatile biomolecules in existence. This year, in particular, we are reminded of the importance of RNA biology as a pandemic caused by an RNA virus has disrupted life worldwide. What are some areas of RNA research that...
  15. mktsgm

    Medical Impact of SARS-2 in areas where SARS-1 was prevalent in 2002-04?

    We know that SARS-CoV-1 affected the world in 2002-04. Many countries including China were affected then. I want to know, how the SARS-CoV-2 is faring now, in the areas which had already been affected by SARS-CoV-1 earlier. Dr. Beda M Stadler the former director of the Institute for Immunology...
  16. M

    B Find triangles with areas that are integers

    It is pretty obvious that all right-angled triangles whose sides are integers will have areas which are also integers. Since either the base or height will be an even number, half base x height will always come out exactly. However, I have only found one non-right-angled triangle where this is...
  17. E

    B Contributions to total energy in different areas of Physics

    Classically, the total energy of a particle/system ##E = T+V##. This is the usage seen in the Schrödinger equation, since E is an eigenvalue of ##\hat{H}## and ##\hat{H} = \hat{T} + \hat{V}##. Then in special relativity, the total energy ##E## of a particle/system becomes the rest energy plus...
  18. S

    Studying Good research areas in pure math

    My name is shabani makwaru from Tanzania i am master student in Pure mathematics at University of Dar es salaam Please can help me in advising which area is good in research in pure math
  19. BillTre

    How to Deal with No Power in Areas of California

    Some areas of California are going to have their power turned off to prevent wild fires like they had last year. I don't live there, so lucky me. Its my understanding that some of the fires were started by wires blowing around and causing sparks in some way. Apparently, the power may be off for...
  20. ChrisXenon

    B Do multiple WiFi devices create areas of high EMR?

    I'm thinking in a typical house or office, areas of constructive interference may exist which aggregate the fields of maybe 10 WiFi devices. Wouldn't this mean the safe power levels stipulated for any single device would be routinely be exceeded? And if so, do we care?
  21. R

    Other Using Databases like Scopus to Find Trends in Physics

    Hello everyone. My question is how can I use databases like scopus to know which areas of physics are the "hot" today or which are still in development.
  22. P

    Admissions Including multiple areas of interest in my SoP for PhD Applications

    I'm starting to write my statement of purpose for Physics and Astronomy graduate schools, but I'm not yet dead set on which field I want to specialize in. Considering this, is it ok for me to write something along the lines of, "I want to either do field 1 with professor I or do field 2 with...
  23. M

    Integral calculation using areas

    Evaluate the integral using the properties of definite integral and interpreting integrals as areas. ##\int_{-1}^2 (1-2x)dx## I need to see there are two areas and these are the same but one is under x-axis the other is above x-axis so the value of the integral is zero. To see this is...
  24. E

    MHB Find the areas of segment in circle

    So far i have. 14) area of sector is πr²/3 = 12π length of chord. that triangle has two sides of 6 and angle of 120º split the triangle in two right triangles with angle of 120/2 = 60 and hyp=6. other (longer) side is: sin 60 = x/6 s = 6 sin 60 = 6(√3/2) = 3√3 third side is s = 3 cos 60 =...
  25. E

    MHB Areas of Plane Figures: 63.0 & 96.0 | Need Help w/ 3 & 4

    I think the answer to number 1 is 63.0 or 60 if rounded to the nearest tenth, and i think number 2 is. 96.0 or 100 if rounded to the nearest tenth. Need help with 3 and 4. I think for 3 i would add 6 and 8 as the vertical diagonal and the horizontal diagonal would be 5 . 6+8=14 so the area...
  26. astroman707

    I What areas of nano/optical research applies to astrophysics?

    I'm an undergraduate and there is a very good optical spectroscopy and nano-materials physics lab at my college. I have FULL access to all equipment and a professor in that field is offering me a ridiculous amount of co-research time during the school year. However, I'm really only interested...
  27. opus

    Right-Endpoint-Sum is negative? Approximating Areas

    Homework Statement Find ##R_4## for ##g(x)=cos(πx)## on ##[0,1]## Homework EquationsThe Attempt at a Solution (i) Since we want ##n=4## even sub intervals between ##x=0## and ##x=1##, intervals are ##[0, \frac{π}{4}]##, ##[\frac{π}{4}, \frac{π}{2}]##, ##[\frac{π}{2}, \frac{3π}{4}]##...
  28. danielFiuza

    How to write this in Set Theory notation?

    Hello Everyone, I am trying to write the intersection of a physical problem in the most compact way. I am not really familiar with Set Theory notation, but I think it has the answer. It is about the intersection of two circular areas: - Area 1: A - Area 2: B If I want to write this in Set...
  29. F

    Experiments about impact of mass in different areas

    There's important experiments showing the equivalence of inertial mass and gravitational mass ( the Eotvos experiments), but I couldn't really find many that show the equivalence of mass (or the stress-energy tensor) in other contexts. These are some of the variants of mass I'm interested in...
  30. B

    MHB Find the ratios of the areas of the four regions

    In square ABCD point M is the midpoint of side CD. Find the ratios of the areas of the four regions (∆MPC, ∆BPC, ∆APB, and quadrilateral APMD) that are formed. Justify your result.
  31. L

    MHB Ratio of areas of squares - Challenging problem

    Hello all, I have encountered a very difficult question in geometry. The question has several parts. I really need your help. I have tried solving the first and second ones, not sure I did it correctly, and certainly don't know how to proceed and what the results means. I would really...
  32. S

    Calculating 3D Cylinder Volumes & Areas With Constants

    Homework Statement Let r be a positive constant. Consider the cylinder x2 + y2 ≤ r2, and let C be the part of the cylinder that satisfies 0 ≤ z ≤ y. (1) Consider the cross section of C by the plane x = t (−r ≤ t ≤ r), and express its area in terms of r, t. (2) Calculate the volume of C...
  33. Hypercube

    Other Availability of funding within different areas of physics

    Hello! I was wondering if any of you researchers on PF are willing to provide some insight in regards to how much funding is available to different areas of physics? I'm happy with just qualitative, personal experience-type answers, like which areas you think have the most, which areas have the...
  34. T

    Biology What type of career areas work with BCIs?

    Hi! I'm interested in learning more about BCIs and the different types of career areas associated with it. I've have a interest in learning about them for commercial use/interaction with everyday objects and was wandering what type of degree/science studies this type of topic/thing... Thanks...
  35. kaliprasad

    MHB Infinite Isosceles Triangles w/ Integer Sides & Areas

    Show that there are infiinite isosceles triangles which have integer sides and integer areas
  36. pairofstrings

    B What area of maths is common to all other areas of maths?

    Hi. I am very curious and fascinated to know what area of mathematics is common to all other areas of mathematics. Thank you.
  37. T

    Studying Is it normal to be a dunce in some areas of physics?

    Maybe "dunce" is the wrong word, but I'm pretty weak on stat mech/thermo. I've had a few courses in them but not to the point where I can understand anything but the most rudimentary basics (each time I took a course in them I happened to have been distracted with other things). I'm a thirdish...
  38. Jarvis88

    Areas of Surfaces of Revolution

    Homework Statement Find the area of the surface generated by revolving the curve y=√x+1, 1≤ x ≤5, about the x-axis. I'm stuck trying to figure out how I can use substitution...if I am even able. I was trying to rewrite 1 as 4(x+1)/4(x+1) but still can't seem to get the right terms to cancel...
  39. L

    Programs Areas of research in Medical Physics

    Hello everyone, I have an upcoming interview with 3 great CAMPEP accredited Medical Physics PhD programs soon. I have went through my BS and MS in physics, with a few courses in Medical Physics (MRI/MRS, Radiation measurement labs, and currently taking Radiation Oncology Undergraduate course)...
  40. L

    Programs Regarding MSc and PhD research areas

    Hi, I'm about to start my MSc in Computer Science studying "Combinatorial Optimization", which was already the theme of my undergraduation thesis. I like the field and the research, but I'd rather prefer working with numerical methods and techniques. The problem is my current university has a...
  41. D

    B Kepler's Law of Areas: Radially Directed Force

    Hi, Just wanted to know how can we attribute the kepler's law of areas to a radially directed force?
  42. M

    Parallel Axis Theorem- Composite Areas (STATICS)

    Homework Statement Consider the beam shown in (Figure 1) . Suppose that a = 15 in. , b = 8 in. , c = 1 in., and d = 4 in. Determine the moment of inertia for the beam's cross-sectional area about the x axis...
  43. W

    Least Exciting Areas of Physics: What Subfield Do You Find the Most Boring?

    Hey guys So I'm sure there are plenty of threads about what people think are the most exciting subfields, but what about the most boring? Which subfield of physics did you enjoy the least learning? I'll begin. In my opinion it has to clearly be scattering theory & everything related (e.g...
  44. FallenApple

    Physics Areas of physics that uses Statistics

    My background is applied mathematics and statistics. A lot of the problems that I have been applying statistical methods towards have been very dry(medical studies, etc). Actually my favorite subject is physics, but I don't have too much experience in it. I am willing to learn. Any suggestions...
  45. Rectifier

    Looking for a software to highlight areas in PDF - Linux

    Hey there! I am looking for a free PDF viewer that is capable of highlighting areas and text and that is available for Debian (Linux). Thanks in advance.
  46. I

    Are different emotions in different areas of limbic system?

    Does anyone know if there has been any data on whether different emotions fire in different areas of the limbic system. Basically, are there "angry" neurons, and "sad" neurons, or to be more direct, neurons that fire exclusively when one is angry, and different neurons that always fire for when...
  47. Kirito123

    Calculating the Area of Rectangles: Solving for Unknown Side Lengths

    Homework Statement 2. Homework Equations A= l x w (area for a rectangle. The Attempt at a Solution I will have to find the area of 1 halve of the roof, since there parallel I only have to find one side. So I know the height is 2 and I also know that the base of the rectangle is 8. I tried...
  48. S

    Courses Interest in Areas of Classical Mechanics

    What are Hamiltonian/Lagrangian Mechanics and how are they different from Newtonian? What are the benefits to studying them and at what year do they generally teach you this at a university? What are the maths required for learning them?
  49. I

    Areas And Distances (Intro. to Definite Integral)

    Hey everyone, Today in my Calculus 1 lecture we covered Areas and Distances, which serves as a prequel to the definite integral in my book. I am confused on some notation the book uses, and I cannot seem to find a clear explanation anywhere that I look. n ∑ f(xi) ΔX ≅ A i=1 First, let me...