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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:The surface area of a hollow cylinder varies jointly with the height and the radius of the cylinder. If a cylinder with radius 13 inches and height 9 inches has a surface area of 106 square inches, what is the surface area of the cylinder with radius 4 inches and height 8 inches?

Answers:would your formula be N = circumference x height proof cut the cylinder open and flop it flat circumference is now length and height is width area is L x W OR circumference times height all the best

Question:1. Wallpaper comes in rolls 6 metres long and 1 metre wide. How much wallpaper is necessary to wallpaper the walls in a bedroom if the bedroom is 4 metres by 5 metres by 2.5 metre?(Don't wallpaper the 1 metre by 2 metre door nor the 1 metre square window.) 2. How many square centimetres of cardboard go into the making of a cardboard box if the box is cube shaped and each of the sides is equal to 60 centimetres? 3. Mr. Fixit is building a laundry room in his basement. It will measure 4 metres by 5 metres by 2.5 metres. If he intends on painting the walls, ceiling and door with two coat of paint, and each litre of paint covers 70m^2, how many litres of paint should he buy? ignore question 1

Answers:I won't solve the problems for you but I'll be happy to give you the formulas for surface area. SA of a Cylinder:: 2*3.14*radius*height+2*3.14*radius^2 SA of Rectangular Solid:: 2(lw)+2(hw)+2(lh)

Question:whats the surface area of a cylinder is 3 is the base and 12 is the height. its just a regular 3-dimensional cylinder.

Answers:= 2*pi*3^2/4 + pi*3*12 = 9pi/2 + 36pi = 81pi/2 sq units

Question:help!!!!!!!!!!!!!!!!!!!!!! i have to solve this math problem about solving a cylinder with the radius of 8 memeters and the heigth is 18 meters it asks me for the surface area. please help me i'll really apreciated.

Answers:Surface Area = Area of 2 circles + Area of rectangle that makes up body of cylinder 1) Area of 2 circles: A = pi * r^2 = pi * 8 ^2 = 64 pi So 2 circles are 128pi 2) Area of rectangle: Length = 18 Width = Circumference of Circle Circumference = 2pi * r = 2pi * 8 = 16pi Area = 18 (16pi) = 288pi 3) Surface Area = 128pi + 288pi = 416pi m^2 Hope that helps :)

From Youtube

Surface Area of a Cylinder :Demonstration on how to solve for the surface area of a cylinder.

Surface Area of a cylinder :WEBSITE: www.teachertube.com How to find the surface area of a cylinder