What is Surface area: Definition and 449 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.

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1. Find the dimensions that will minimize the surface area of a Rectangle

My interest is on number 11. In my approach; ##v= xyz## ##1000=xyz## ##z= \dfrac{1000}{xy}## Surface area: ##f(x,y)= 2( xy+yz+xz)## ##f(x,y)= 2\left( xy+\dfrac{1000}{x} + \dfrac{1000}{y}\right)## ##f_{x} = 2y -\dfrac{2000}{x^2} = 0##...
2. Find the partial diameter error of the surface area of cylinder

(ΔSA/ΔD) = 2πHΔD Something is wrong I guess as I get wrong value.
3. I Surface area and length (percent increase relationship)

Hi all, While calculating the surface area for an object, I was told the below statement. However, I am not sure is this correct, please can someone help me to explain this with an example? Is the below statement always true? The surface area % increase should be in line or less than the %...
4. Find the surface area of the given solid

My question is on how did they determine the limits of integration i.e ##2## and ##3## as highlighted? Thanks
5. B Oriented Surfaces & Surface Area: Investigating the Impact

I usually think of a sphere as the set of all points ##P_x##, that have the identical distance r to some point ##C## which is the center of the sphere. I calculate the surface area ##A## of the sphere as $$A=4 \pi (C P_x)^2$$ However, what happens if I think of the distance between the points C...
6. Voltage and electrode surface area

Hi I am looking to find the equation that determines the minimum (and if possible maximum that might damage the electrode) voltage that starts the electrolysis process for a given area of a graphite electrode in a brine solution medium (lets say 30%) at equilibrium state. Also how does the...
7. Spirit evaporating from a bowl

Problem statement : I draw the problem statement above. I hope I am correct in inferring that the bowl is hemispherical. Attempt : I could not attempt to the solve the problem. We are given that the rate of change (decrease) in volume is proportional to the surface area ...
8. Surface area of a sphere ##= \pi * (a^2+b^2+c^2+d^2)##

I am not very good at proofs. The only thing I have come up with is the following regularity. However, I am not sure how this can be related to the above problem. Given a sphere ##S_a## with a center ##C## and a diameter of ##a##. I can now construct a line segment ##b## with the endpoints...
9. I Calculating the surface area of a sphere using dA

Below is an image to calculate the surface area of a sphere using dA. I can see how ##rcos\theta d\phi## works, but I don't understand how that side can't just be ##rd\phi## with a slanted circle representing the arc length. The second part I don't understand is why it is integrated from...
10. I don't understand this problem -- Surface area of a section of a solid sphere

Sorry i don't understand English very well , if someone want to explain to me this problem?
11. Find the surface area of the water in the given prism

My query in only on the highlighted part...c.ii. Find the question below; Find the markscheme here part c(ii) does not seem correct as i have; ##A_1=0.5 ×(0.65+0.84)0.3 ×2=0.447m^2## ##A_2 = 0.65 ×1.6=1.04m^2## ##A_3 = (0.3146 × 1.6)2=1.00672m^2## Total surface area =...
12. I Quasi-Static Change of Event Horizon Area

Let ##\mathscr{H}## be a constant-##v## cross-section of the event horizon (area ##A##). The expansion is the fractional rate of change of the surface element, ##\theta = \frac{1}{\delta S} \frac{d(\delta S)}{dv}##. The problem asks to prove the formula ##\frac{dA}{dv} = \frac{8\pi}{\kappa}...
13. Effect of Surface Area on the Drag Coefficient of a Parachute

Summary:: Does the surface area of a parachute affect its drag coefficient? If so, how? I have been trying to figure out the effect of surface area on the drag coefficient of a parachute. I have designed a lab in which parachutes of different surface areas are dropped and the terminal velocity...
14. B How to determine force based on pressure

Hello In a pharmaceutical lab,a certain pressure is applied to IV bags using the equipment shown below.We need to calculate the force acting on the bags based on the applied pressure.I know the formula is F=PxA.But i am not sure what the surface area is.Should i take the whole surface area of...
15. MHB Calculate Volume & Surface Area of a Cylinder Without Lid

how to find volume and surface area of this without using the upper lid
16. What is the PV panel surface area?

Hello everyone, I am trying to do some calculations for the energy output of a solar farm that I am designing as my dissertation. However, when I trie to calculate the following formula: Wp = ηpvGBA from Equation (11) above, where: ηpv is module efficiency (18.4%) GB is solar irradiance (3.8)...
17. Rates of change: surface area and volume of a sphere

The surface area of the sphere is 4πr^2. dr/dt is given as 3cm^-1. dS/dt=dS/dr*dr/dt Differentiating 4πr^2 is dS/dr= 8πr dS/dt=8πr*3 dS/dt=24πr Given that r=5 dS/dt=24π*5=120 π The volume of the sphere is 4/3πr^3, differentiating which is dV/dr=4πr^2 dV/dt=dV/dr*dr/dt dV/dt= 4πr^2*3...
18. I The surface area of an oblate ellipsoid

In "An Introduction to Nuclear Physics by W. N. Cottingham, D. A. Greenwood" for the surface area of an oblate ellipsoid, the following equation is written for small values of ε : The book has said this without proof. I found the following formula for the desired shape: No matter how hard I...
19. MHB [ASK] Minimum Surface Area

The volume of a cuboid box with a square base is 2 litres. The production cost per unit of its top and its bottom is twice the production cost per unit of its lateral sides. Suppose the side length of its base is x and the height of the cuboid is h. The minimum production cost is reached when...
20. I Calculating Surface Area of Schwarzschild Black Hole w/Weyl Coordinates

Recently, I was tasked to find the surface area of the Schwarzschild Black Hole. I have managed to do so using spherical and prolate spheroidal coordinates. However, my lecturer insists on only using Weyl canonical coordinates to directly calculate the surface area. The apparent problem arises...
21. I Why volume is conserved but not the surface area?

A water drop of radius ##10^{-2}## m is broken into 1000 equal droplets. Calculate the gain in surface energy. Surface Tension of water is ##0.075 ~N/m##. So, for the solution of the above problem we need to know how much surface area (combining all 1000 droplets) have increased from the...
22. I Partial Surface Area of a Tube

Hi all, I hope this is the correct place to post this. Below is a section of a pipe. The pipe has a radius of 0.848 m. For this example, assume the pipe is buried below ground but a section of it remains exposed. The centre of the pipe is buried 0.590 mbelow the ground. If we assume the pipe...
23. B Area of a Circle in an Electron's Hydrogen Atom

My textbook says "A is the area of the circle enclosed by the current" (produced by an electron in a hydrogen atom), A = ##\pi r^2 \sin(\theta)^2##. I don't understand where the ##\sin(\theta)^2## comes from.
24. MHB Volume and surface area of "One World Trade Center"

Hi to everyone, do you know the "One World Trade Center"? Well, I've to calculate two things about it: -The volume, according to its particular shape -The surface of the glass plates which cover the whole structure Searching on internet i found two dimensions: 1) Total height without...
25. Thermal expansion (surface area)

Hi all, For area expansion, I know the equation goes like: Hence, my answer to part i is ##\begin{aligned}A=A_{0}\left( 1+2\alpha \Delta T\right) \\ = 52\left( 1+2*24\times 10^{-6}\right) \left( 100\right) \\ =52.2496cm^{2}\end{aligned} ## Now I am unsure how to proceed with part 2 in this...
26. Why does an increase in surface area lead to a reduction in pressure?

From what I read on the internet I found that increase in surface area that is in contact is offset by the reduction in pressure. What exactly does it mean? This is what I understood from the it (but my understanding might be absurd :-p): does reduction in pressure mean that the "hills" or...
27. Element of surface area in spherical coordinates

r,θ,ϕ For integration over the ##x y plane## the area element in polar coordinates is obviously ##r d \phi dr ## I can also easily see ,geometrically, how an area element on a sphere is ##r^2 sin\theta d\phi ## And I can verify these two cases with the Jacobian matrix. So that's where I'm at...
28. Calculating the surface area of a solid of rotation

For some reason I have become very unsure but my gut feeling says i can calculate y=(1-x^(a/b))^(d/c) I already know the formula for calculating the volume. but can transfer the whole thing as a function of y(x) and take the integral then as a single integral?
29. I Strain of of a bubble's surface interface during vibrations

Imagine a bubble vibrating in air. Because it vibrates, it's interfacial area increases, thus new molecules are added and removed from the surface as it vibrates. Consider a molecule is initially at position X_0 at the interface, and over a certain amount of time molecules squeeze and disappear...
30. Earth-like planet with 2x surface area, but same surface gravity?

How feasible is such a planet? Can the planet still be dense enough to be rocky and not gaseous?
31. Calculus of Variations, Isoperimetric, given surface area max volume

My volume integral is... $$\pi\int y^2 dx$$ My surface area integral is... $$2\pi\int y \sqrt {1+x'^2} dy$$I'm fairly sure the variable of integration on my volume and surface area integrals has to be the same, is that right? But when I change the variable in the surface area integral to...
32. Surface area to volume ratio

In a human body we have two relatively similar structures -pulmonary...
33. Checking divergence theorem inside a cylinder and under a paraboloid

I am checking the divergence theorem for the vector field: $$v = 9y\hat{i} + 9xy\hat{j} -6z\hat{k}$$ The region is inside the cylinder ##x^2 + y^2 = 4## and between ##z = 0## and ##z = x^2 + y^2## This is my set up for the integral of the derivative (##\nabla \cdot v##) over the region...
34. Understanding the argument of the surface area integral

Homework Statement Find ##\iint_S ydS##, where ##s## is the part of the cone ##z = \sqrt{2(x^2 + y^2)}## that lies below the plane ##z = 1 + y## Homework EquationsThe Attempt at a Solution [/B] I have already posted this question on MSE...
35. Understanding why we compute surface area as we do

Homework Statement Homework Equations The Attempt at a Solution [/B] The solution to this problem is known. I want to use this exercise as a model to understand how to proceed when calculating the surface area of a geometric figure. Question: 1) Why do we differentiate with...
36. Minimal surface area for a fixed volume

Homework Statement a hut has to side walls a roof and back wall. its front is open. its total volume is 120m^3 fdetermine the miniumal surface area necessary for a sheet to be put over it Homework EquationsThe Attempt at a Solution Attempt 2 V=xyz=120 z=120/xy s = 2yz + xz + xy s = 2y(120/xy)...
37. Minimizing the surface area of an open box

Homework Statement An open topped box with a square base has the capacity of ##32m^2##. Find the dimensions that will minimize the surface area of the box. Homework EquationsThe Attempt at a Solution I was told these are the dimensions, but I can't picture them in my head at all...
38. Surface area of a shifted sphere in spherical coordinates

Homework Statement find the surface area of a sphere shifted R in the z direction using spherical coordinate system. Homework Equations $$S= \int\int \rho^2 sin(\theta) d\theta d\phi$$ $$x^2+y^2+(z-R)^2=R^2$$ The Attempt at a Solution I tried to use the sphere equation mentioned above and...
39. I Is There a Curve For a Material's Surface Area vs. UTS

I drew a diagram in order to help figure out something for a tabletop game I'm putting together. My question is about the physics of materials, and is not directly about the fictitious psychic/magic abilities in my game world. I drew a diagram consisting of dots representing particles and...
40. Vector Calculus, setting up surface area integral.

The question goes like: find the SA of the portion S of the cone z^2 =x^2 +y^2 where z>=0 contained within the cylinder y^2+z^2<=49 this is my attempt using the formula for SA, I could switch to parametric eqns, but even then I'd have hard time setting up limits of integration.
41. I Glass -- how to increase the surface area

I need to increase the surface area of the glass which will be used in a solar still with the intention of keeping the glass as cool as possible. My first thought was bubble wrap because it's transparent and I thought it would not interfere with the light but then I remembered it is a good...
42. I Gabriel's Horn, Inside vs. Outside Surface area

Consider Gabriel's Horn, the mathematical object formed by a surface of revolution of the curve x= 1/x from x=1 to infinity. It is known that one can fill the horn with a volume of Pi cubic units of paint but it would take an infinite amount to paint the surface. I think they usually mean the...
43. I How to measure the surface area of an arbitrary 3D object

I'm looking for an easy way to get the surface area of an arbitrary shaped 3D object. Getting the volume is easy by water displacement. What about area? Any neat tricks? We know different shapes can have the same volume and thus different surface areas so it's not a trivial problem. The purpose...
44. 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...
45. Surface area of a cylinder- finding unknown variable

Homework Statement The surface area, A, of a cylinder with height, h, and radius, r, is given by the equation ##A=2πrh+2πr^2##. A company makes soup cans by using 32π square inches of aluminum sheet for each can. If the height of the can is 6 inches, find the radius of the can. Homework...
46. Surface and volume element

Homework Statement Show that the surface and volume element on a deformed sphere are \sigma = \frac{\rho^2 \sin \theta}{\cos \gamma} d\phi d\theta, dV = \rho^3 \sin \theta d\phi d\theta, if \gamma is the angle between normal vector and radius vector. Homework Equations n\cdot r = \cos...
47. Gas adsorption for (BET) surface area of graphene oxide based thin films

I am using a Micromeretics ASAP2020 machine. The sample is ~25mg of reduced graphene oxide based film. The problem is I am getting adsorption isotherms showing negative volumes of adsorbate (N2) entering the tube as P/P0 increases. The sample was degassed prior to testing at 250°C for 3...
48. I Surface area of inner and outer halves of torus

A torus with major radius, ##R##, and minor radius, ##r##, has a total surface area given by ##4\pi^2 Rr##. If one slices the torus on its midline (i.e. at a line on a poloidal angle of ##-\pi/2## and ##\pi/2##), I was told the inner half of the torus has a smaller surface area than the outer...
49. J

I Evaporation under vacuum; surface area

I run a vacuum distillation unit that is used to distill ethylene glycol. The old glycol is subjected to 25Hg of vacuum at a temp of 275 degrees F. There are burner tubes submerged in the glycol to heat it up. The vessel has a diameter of 36 inches and is 124 inches long. We fill the vessel...
50. 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...