What is Surface area: Definition and 450 Discussions

The surface area of a solid object is a measure of the total area that the surface of the object occupies. The mathematical definition of surface area in the presence of curved surfaces is considerably more involved than the definition of arc length of one-dimensional curves, or of the surface area for polyhedra (i.e., objects with flat polygonal faces), for which the surface area is the sum of the areas of its faces. Smooth surfaces, such as a sphere, are assigned surface area using their representation as parametric surfaces. This definition of surface area is based on methods of infinitesimal calculus and involves partial derivatives and double integration.
A general definition of surface area was sought by Henri Lebesgue and Hermann Minkowski at the turn of the twentieth century. Their work led to the development of geometric measure theory, which studies various notions of surface area for irregular objects of any dimension. An important example is the Minkowski content of a surface.

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

    How do i choose the Surface area vector in different cases?

    Homework Statement Suppose we have a Electic field, E (vector) = kr2r(vector). Find the total charge contained in the sphere of radius R centered in the origin. solution- here E can also be written as kr3r∧, where r∧ is the unit vector this is the given question , so obiously the best way to...
  2. C

    I Surface Area of Volume of Revolution

    The problem is, find the surface area of the volume of revolution generated by rotating the curve y=e2x between x=0 and x=2 about the x-axis. Here's what I have so far... SA=∫y√(1+y2)dx =∫e2x√(1+4e4x)dx and from here I'm not really sure what to do. Any help would be appreciated.
  3. Kaneki123

    B A Question about Effect of Surface Area on Loudness

    Okay... I have a question about "effect of surface area on loudness".I am going to upload a diagram along this post.It shows two scenarios. The "red thing" is a barrier, which allows absolutely no air contact between the speaker and the listener. So the only way the sound can travel outside is...
  4. L

    Surface area of a circular cylinder cut by a slanted plane

    Homework Statement Cylinder : x^2 + y^2 = 1 Plane that intersects above cylinder: y + z = 2 What is the surface area of the sides of this cylinder? Homework Equations dS= R1 d@ dz @ is from 0 to 2 pi z is from 0 to 2 - y dS=(Zx^2 + Zy^2 + 1)^.5 dA Where Z = 2 - yThe Attempt at a Solution I...
  5. K

    Finding Depth of Water Loss Using Surface Area

    Homework Statement A city with 30,000 residents lies next a lake which is their only source of water. The average family of four in this city uses 1100 liters of water per day. Neglecting rain and evaporation, how much depth would the lake lose per year if it covered 40 square kilometers...
  6. Anachronist

    Intuitively, which of these coroplast fins has less drag?

    Here's a crude model rocket fin made out of a material called Coroplast (corrugated plastic): And here's the exact same fin with the flutes cut out from all around the edges, allowing the edges to be squeezed shut and sealed with tape: It may look like there are small openings, but...
  7. Hiero

    How the surface area of a cone changes; special relativity

    Homework Statement A cone has half angle θ0 and lateral surface area S0 in the frame in which the cone is at rest. If someone moves at relative speed β=v/c along the cones symmetry axis, what surface area will they see for the cone? Homework Equations I believe the lateral surface area of a...
  8. Kevin Shen

    Pressure, surface area and force in tire

    Assuming that the temperature in the tire suddenly rises. My question is that would that cause the contact surface area between the tires and the floor increase, decrease, or stay the same? (given that the volume of air in the tires is a constant and cannot be changed) Also, why?
  9. M

    MHB How Much Netting Does Rita Need to Cover a Rectangular Area?

    Rita wants to cover a roughly rectangular area with netting. The height is 9 feet (but one side is along a solid fence, so could be 4 feet), two sides are each 6 feet, and the other side is 5_1/2 feet. How much netting does she need? Netting comes as a rectangular or square piece. My Work:Let A...
  10. T

    B About the surface area of a prolate ellipsoid

    Is there any limit for which we can approximately write the surface area of a prolate ellipsoid to be 4piA*B comparing with the spherical 4piR*R??
  11. A

    B How to find the appropriate area in Gauss' law

    Knowing that Gauss' law states that the closed integral of e * dA = q(enclosed)/e naut, how would you find exactly what A is in any given problem? I know it varies from situation to situation depending on the geometry of the charge. For instance, I know that for an infinite wire/line of...
  12. T

    Finding Volume and Surface Area of a Banana Using Calculus

    Homework Statement We are given a Banana, and asked to find the volume and surface area of the function, using calculus. So far, we have learned elementary calculus (derivatives, limits, and integrals) as well as volumes of revolutions. We traced the banana on graph paper, plotted points on the...
  13. adsorption

    I Anyone knows V-Sorb 2800 BET surface area analyzer principle

    we interest one V-Sorb 2800 BET surface area analyzer, using physical adsorption principle to test particles surface area data, if anyone knows this analyzer principle?
  14. Z

    Formula for Total Surface Area

    Homework Statement A 5-foot long cylindrical pipe has an inner diameter of 6 feet and an outer diameter of 8 feet. If the total surface area (inside and out, including the ends) is k*PI , what is the value of k? Homework Equations In my view the formula should be: 2* PI * radius * h + 2*PI *...
  15. D

    I How do I calculate the surface area of a rotated curve?

    How do I find the surface area of a f(x) rotated around the y axis?
  16. C

    Surface area bounded by 2 different planes

    Homework Statement Find the surface area of portion of plane x + y + z = 3 that lies above the disc (x^2) + (y^2) < 2 in the first octant ... Homework EquationsThe Attempt at a Solution Here's the solution provided by the author ... I think it's wrong ... I think it should be the green...
  17. T

    Frictional force and surface area in contact

    I've read that the surface area of an object in contact with the ground doesn't not affect the frictional force acting on it as it is pushed forward. I kinda understand what is explained but I find it difficult to reconcile with what happens in real life... Don't wheels reduce the surface...
  18. C

    I Why is the limit of θ from 0 to π in the formula for surface area of a sphere?

    I found this on the Internet . The formula is Surface Area = R^2 \displaystyle \int _0 ^ {2 \pi} \int _{0}^{\pi} \sin \theta d \theta d \phi I'm wondering why the limit of θ is from 0 to π only ? why not from 0 to 2π ? Because it's a perfect sphere...
  19. Oannes

    Finding Surface Area in square feet with Volume & Thickness

    Homework Statement How large a surface area in units of square feet will 1 gallon of paint cover if we apply a coat of paint that is 0.1cm thick? Homework Equations Since Volume is L * W * H and we can assume the object is square besides the height which in this case will be the thickness. So...
  20. 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...
  21. B

    Automotive Single actuator Wiper design to wipe a surface area

    I'm given this problem to solve with the assumption as stated above. The answers need not be logical as long as the linkage mechanism can be simulated. I've attempted the question using Mercedes Mono Wiper mechanism but only manage to cover 41% given the width of the wiper. May I ask if there...
  22. G

    Determining the surface area on a 5 sided lunar esque shape

    Homework Statement I'm trying to figure out the surface area on a 5 sided shape where the sides can all be modeled by "lunes". The shape will end up looking like a banana peel. We are modeling the sides of the shape as lunes with varying angles on a sphere of radius 3 inches. I'm trying to...
  23. Nemo1

    MHB Surface area of sphere increase at a constant rate

    Hi Community, I have this tutorial question. When I look at the first question (a) I think it is FALSE as the surface area would not increase at the same rate as the radius. For the second question I am not sure if I am interpreting it correctly. If r=\sqrt{\frac{Ct}{4\varPi}+2} where Ct is...
  24. Destroxia

    Surface Area of Parameterized Region

    Homework Statement Find the surface area S of the portion of the hyperbolic paraboloid: ## r(u,v) = \langle (u+v),(u-v),uv \rangle ## for which: ## u^2 + v^2 <= 225 ##Homework Equations (Surface Area for Parameterized Region:) ##\int \int ||\frac {\partial r} {\partial u} \times \frac...
  25. Thales Costa

    I Parameterize an offset ellipse and calculate the surface area

    I'm given that: S is the surface z =√(x² + y²) and (x − 2)² + 4y² ≤ 1 I tried parametrizing it using polar coordinates setting x = 2 + rcos(θ) y = 2rsin(θ) 0≤θ≤2π, 0≤r≤1 But I'm not getting the ellipse that the original equation for the domain describes So far I've tried dividing everything...
  26. B

    Measuring Alkyl Chain Cross-Sectional Area for Gibbs Adsorption

    1. Surfactant molecule is made from water-loving head and grease-loving tail (Figure 1). My question: How do we measure the cross-sectional area of the alkyl chain of surfactant? Do we measure it vertically (refer to GREEN DOUBLE ARROWS of Figure 1) or horizontally (refer to RED DOUBLE ARROWS of...
  27. M

    Surface area of a spherical cap by integration

    hi guys, i have a question. i saw this picture, and i don't really understand how they derived with the formula. The aim is basically to find the formula for the surface area of a spherical cap. why do you differentiate the x=sqrt(rˆ2-yˆ2)? how does that help to find the surface? and then...
  28. T

    Silly theoretical area question

    Just humour me, if you had an infinitely thin cone, would the surface area inside the cone be the same surface area on the outside of the cone? It must be right? Is there a formula for the surface areas of the inside and outside of a cone WITH thickness?
  29. T

    Surface area of a sphere with calculus and integrals

    Homework Statement How do I find the surface area of a sphere (r=15) with integrals. Homework Equations Surface area for cylinder and sphere A=4*pi*r2. The Attempt at a Solution I draw the graph for y=f(x)=√(152-x2). A circle for for positive y values which I rotate. I will create infinite...
  30. M

    Sphere Volume to Surface Area, Why not for Cone?

    Homework Statement Wikipedia tells me that I can obtain the surface area of a sphere by realizing that the volume of a sphere is equivalent to the infinite sum of the surface areas of hollow, nested spheres, sort of like little Russian dolls. That makes sense, and then differentiating both...
  31. M

    Calculating the Surface area of a molecule

    Hi, So for a piece of maths coursework I am thinking of trying to calculate the surface area of an atom or molecule. I do not know whether it would be viable because there isn't a clear boundary for an atom/molecule due to the electron clouds. However I wanted to know if I could do something...
  32. S

    Minimizing surface area of a shaker

    Hi, I have an mathematics assignment to do, and I wonder if the topic I have chosen is doable for me. I want to minimize the surface area of a cobbler cocktail shaker, and until now my plan was to get the curve equation for the side of it, and get the area equation from surface of revolution...
  33. alphaj

    Relating Newton's Force to Black Holes (Calculation?)

    Homework Statement Muscle can be torn apart by a force of 100,000 N applied across an area of 1 m2. A 10 cm2 muscle therefore will be torn by a force of 100 N. If a student of average size were being lowered into a black hole of 1 solar mass, at about what distance from the hole's center will...
  34. Legolaz

    How Do You Calculate the Surface Area of a Helix?

    Hi Forum, Just wanted to ask how to compute surface area of a helix x =cos t, y =sin t, z=t by integration Thanks in advance.
  35. W

    Why do we use racing slicks if surface area =/= friction?

    I understand how surface area does not come into the equation governing the resistance an object has to sliding. It leads one to believe that all tires would perform equally in dry conditions. Why do people then say that bald tires give better grip?
  36. D

    Surface area of a cone

    Homework Statement Determine a simplified, factorised expression, in terms of the radius (r), for the surface area of a cone where diameter (D) = perpendicular height (h) Homework Equations A = πr (r + √(h^2 + r^2)) The Attempt at a Solution h=D=2r A = πr (r + √(2r^2 + r^2)) A/π = r (r +...
  37. N

    Li ion battery electrode surface area

    What effect (if any) would changing the surface area of electrodes in a lithium ion battery have? Would it allow for faster battery charging? Faster discharge rate? Thanks in advance
  38. Stephanus

    Does a black hole have a surface area?

    [Moderator's note: Spin-off from another thread.] I'm sorry, does black hole have surface area? Did you mean the sphere defined by Schwarzschild Radius?
  39. D

    Question about surface area of faraday cage

    Hi, I am making a faraday cage to be used in reactive ion etching of silicon. I was wondering if the size of the cage, or the surface area has any impact on it's effectiveness or how it works? I know the size of the mesh and the material I use, as well as whether or not it is grounded all...
  40. S

    Need help finding the minimum surface area of a cylinder?

    Member warned about posting without showing an effort 1. Homework Statement 'Using the graphing function on your Graphics Calculator, or otherwise, determine the radius for a minimum Surface Area.' I HAVE NO IDEA WHAT TO DO? Homework EquationsThe Attempt at a Solution
  41. P

    Surface Area of y=sin^2(x)+x^2 from 0 to 1 about x axis

    Homework Statement Set up a definite integral for the surface area generated by rotating the curve ##y= \sin^2x+x^2## from ##x=0## to ##x=1## about the a-axis. Homework Equations Surface Area about x axis=##2 \pi y \cdot ds ## The Attempt at a Solution I found ##\dfrac{dy}{dx} = 2 \sin x\cos...
  42. Tarpie

    Surface area of revolution about y

    Homework Statement [/B] Find the surface area obtained by rotating the curve y = x^2/4 - ln(x)/2 1 \leq x \leq 2 Homework Equations 2π \int f(x)\ \sqrt{1+(f'x)^2} dx The Attempt at a Solution I can't seem to isolate for x in terms of y. I raised both sides to e and separated the exponents...
  43. Tarpie

    How to Rotate a Surface Area about the Y-Axis?

    Greetings, y=x2/4 - ln(x)/2 from 1=<x<=2 rotated about the y-axis. I did the equation rotating about the x-axis via 2pi* integral (f(x)*sqrt(1+f'(x)^2)) dx with dy/dx = x/2 - 1/2x but the question calls for rotation about y and i can't seem to rearrange the equation to isolate for...
  44. T

    Increase Catalyst Surface Area & Reaction Rate: Experiment

    Hi there, I am doing an experiment in increasing the mass of MnO2 when it is added to H2O2 decomposition, and I'm measuring the rate of temperature change. I chose increments of 0.050, 0.100, 0.150, 0.200, and 0.250 g to put into H2O2 when it is decomposed, i.e. the MnO2 is a catalyst. I...
  45. A

    Heat transfer and surface area of a pipe

    So a friend of mine is living in an apartment with central heating but the radiators are removed, so there are just the pipes. He seems to think that if you cut aluminium jar caps and cover the pipes with them, it will increase the surface area of the pipe and therefore heat the room more. The...
  46. 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?
  47. 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...
  48. J

    Aluminum structure modifications for use near a very hot fire

    I'm building an frame out of solid 5/8 inch thick aluminum square bars. this frame will be around a fire temps close to melting points but there will be insulation in make sure it won't melt. I was thinking about drilling holes in the free space of the aluminum to reduce it's weight and if you...
  49. 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...
  50. 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...
Back
Top