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difference between lateral area and surface area

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From Wikipedia

Surface area

Surface area is the measure of how much exposed area a solid object has, expressed in square units. Mathematical description of the surface area is considerably more involved than the definition of arc length of a curve. For polyhedra (objects with flat polygonal faces) 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 the surface area is based on methods of infinitesimal calculus and involves partial derivatives and double integration.

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.

Definition of surface area

While areas of many simple surfaces have been known since antiquity, a rigorous mathematical definition of area requires a lot of care. Surface area is an assignment

S \mapsto A(S)

of a positive real number to a certain class of surfaces that satisfies several natural requirements. The most fundamental property of the surface area is its additivity: the area of the whole is the sum of the areas of the parts. More rigorously, if a surface S is a union of finitely many pieces S1, …, Sr which do not overlap except at their boundaries then

A(S) = A(S_1) + \cdots + A(S_r).

Surface areas of flat polygonal shapes must agree with their geometrically defined area. Since surface area is a geometric notion, areas of congruent surfaces must be the same and area must depend only on the shape of the surface, but not on its position and orientation in space. This means that surface area is invariant under the group of Euclidean motions. These properties uniquely characterize surface area for a wide class of geometric surfaces called piecewise smooth. Such surfaces consist of finitely many pieces that can be represented in the parametric form

S_D: \vec{r}=\vec{r}(u,v), \quad (u,v)\in D

with continuously differentiable function \vec{r}. The area of an individual piece is defined by the formula

A(S_D) = \iint_D\left |\vec{r}_u\times\vec{r}_v\right | \, du \, dv.

Thus the area of SD is obtained by integrating the length of the normal vector \vec{r}_u\times\vec{r}_v to the surface over the appropriate region D in the parametric uv plane. The area of the whole surface is then obtained by adding together the areas of the pieces, using additivity of surface area. The main formula can be specialized to different classes of surfaces, giving, in particular, formulas for areas of graphs z = f(x,y) and surfaces of revolution.

One of the subtleties of surface area, as compared to arc length of curves, is that surface area cannot be defined simply as the limit of areas of polyhedral shapes approximating a given smooth surface. It was demonstrated by Hermann Schwarz that already for the cylinder, different choices of approximating flat surfaces can lead to different limiting values of the area.

Various approaches to general definition of surface area were developed in the late nineteenth and the early twentieth century by Henri Lebesgue and Hermann Minkowski. While for piecewise smooth surfaces there is a unique natural notion of surface area, if a surface is very irregular, or rough, then it may not be possible to assign any area at all to it. A typical example is given by a surface with spikes spread throughout in a dense fashion. Many surfaces of this type occur in the theory of fractals. Extensions of the notion of area which partially fulfill its function and may be defined even for very badly irregular surfaces are studied in the geometric measure theory. A specific example of such an extension is the Minkowski content of a surface.

Common formulas

In chemistry

Surface area is important in chemical kinetics. Increasing the surface area of a substance generally increases the rate of a chemical reaction. For example, iron in a fine powder will combust, while in solid blocks it is stable enough to use in structures. For different applications a minimal or maximal surface area may be desired.

In biology

The surface area of an organism is important in several considerations, such as regulation of body temperature and digestion. Animals use their teeth to grind food down into smaller particles, increasing the surface area available for digestion. The epithelial tissue lining the digestive tract contains microvilli, greatly increasing the area available for absorption. Elephants have large ears, allowing them to regulate their own body temperature. In other instances, animals will need to minimize surface area; for example, people will fold their arms over their chest when cold to minimize heat loss.

The From Yahoo Answers

Question:like one of them is just the floor right? and the other is the ceiling,walls, and floor which is which and could you explain? I really dont remember anything about how to find area. Thanks the TAKS test is tomorrow! wish me luck :P

Answers:Surface area is used for 3 dimensional objects like cans and balls and ice cream cones. Area is usuall used for plane objects like the area in a circle or the area in a square. Ex: Put a page from the newspaper flat on the floor and figure out how much SURFACE AREA of the floor is covered. What is the AREA of the paper? What is the SURFACE AREA of a ball? What is the AREA of the shadow from a billboard in the sun? Hope the examples are helpful.

Question:Here's the example I'm given: http://learn.flvs.net/webdav/educator_math2_v5/module10/imagmod10/10_03a_03.gif To find the surface area, visualize the net for this figure. The net consists of a large rectangle (the lateral faces) with sides of 12 cm and the perimeter of the base (5cm +5 cm +8 cm +8 cm = 26 cm). The area of the lateral faces would be 12 cm X 26 cm = 312 square cm The bases of the prism are rectangles with length and height of 8 cm and 5 cm. The area of this would be 8 cm X 5 cm = 40 square cm Adding the lateral area and 2 bases together will reveal the surface area. 312 sq cm + 2(40 sq cm) = 392 sq cm I dont really get it. Can somebody help me find a way to remember this, and explain it? Thanks. http://i31.tinypic.com/153ty7l.png

Answers:Just so you know, your link led to a complaint page because the system with the example expects a "cookie" which, of course, was only set on your computer. But this example is fairly straightforward, though the term "net" is a little odd. What they are doing is "unfolding" the sides of the prism to calculate the surface area. The four sides ("lateral faces") unfold into one large rectangle, and that leaves the top and bottom ("bases") to be added in. So they calculate the area of the lateral sides (height times the perimeter of the base = 312) and then add the area of the two bases, each being length x width (which gives the 2 * 40). Here's another, equivalent way to look at it: take the three dimensions of the rectangular as x, y, and z. For each pair of measurements, there will be two opposite faces that consist of rectangles with that pair of dimensions. So the area is 2(xy + xz + yz). In the example, x=5, y=8, z=12 (or you can assign them in any other order and it will come out the same). So the surface area is 2 (5*8 + 5*12 + 8*12) = 2 (40 + 60 + 96) = 2 * 196 = 392 Their approach, unfolding the sides and then adding the top and bottom, just collects the areas of the sides in a different order. You're still adding up six rectangles; it's just that they've stuck four of them together in one step.

Question:Could someone explain (with steps) how to solve a problem similar to this one (I have tried it twice and gotten different answers both times; also any equations needed would help. Thank you. Sphere 1 has a surface area of A1 and a volume of V1, and sphere two has a surface area of A2 and volume of V2. IF the radius of sphere 2 is double the radius of sphere 1, what is the ratio of (1) the areas A2/A1 and (2) the volumes of V2/V1.

Answers:So let r be the radius of sphere 1 let 2r be the radius of sphere 2 (duh, sort of). A1 = 4 r^2 A2 = 4 (2r)^2 = 4 *4*r^2 = 16 r^2 A2/A1 = (16 r^2)/(4 r^2) = 4 V1 = (3/4) r^3 V2 = (3/4) (2r)^3 = 6 r^3 V2/V1 = (6 r^3)((3/4) r^3) = 8

Question:please write a paragraph how to describe it..

Answers:the volume of a figure is the amount of space occupied or that can be occupied inside of the object. the surface area however is what is showing on the surface of the object. the surface area does not include the inside of a figure only the outside surface.

From Youtube

Venturi Effect Fluid Turbine .58 square meter lateral surface area :Venturi Effect Fluid Turbine lifting 1.1kilos @ 5m/s wind velocity, 3.3 n/m of torque. CFD analysis confirms 55% Betz limit.

Surface Area - Octagon :A maths lesson starter for KS3 students showing the step-by-step process of working out the area of an octagon. Using an octagonal shape found in a shopping mall, the video demonstrates how to work out its area using a variety of methods, some of which can also be applied to any regular polygon. This engaging way of presenting different methods for calculating the area of shapes is suitable for use with Years 7 to 9.