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Woldemar Tikhonov
Woldemar Tikhonov

Surface Area [PORTABLE]

The surface area of a solid object is a measure of the total area that the surface of the object occupies.[1] 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.

surface area

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.

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 the 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

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; this example is known as the Schwarz lantern.[2][3]

Various approaches to a 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 an area to it at all. A typical example is given by a surface with spikes spread throughout in a dense fashion. Many surfaces of this type occur in the study 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 geometric measure theory. A specific example of such an extension is the Minkowski content of the surface.

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.

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 surface area of a solid is a measure of the total area occupied by the surface of an object. All of the objects addressed in this calculator are described in more detail on the Volume Calculator and Area Calculator pages. As such, this calculator will focus on the equations for calculating the surface area of the objects and the use of these equations. Please refer to the aforementioned calculators for more detail on each individual object.

Xael doesn't like sharing her chocolate truffles with anyone. When she receives a box of Lindt truffles, she proceeds to calculate the surface area of each truffle in order to determine the total surface area she has to lick to decrease the probability that anyone will try to eat her truffles. Given that each truffle has a radius of 0.325 inches:

The surface area of a circular cone can be calculated by summing the surface area of each of its individual components. The "base SA" refers to the circle that comprises the base in a closed circular cone, while the lateral SA refers to the rest of the area of the cone between the base and its apex. The equations to calculate each, as well as the total SA of a closed circular cone are shown below:

Athena has recently taken an interest in Southeast Asian culture, and is particularly fascinated by the conical hat, typically referred to as a "rice hat," which is commonly used in a number of southeast Asian countries. She decides to make one of her own, and being a very practical person not mired in sentimentality, retrieves her mother's wedding dress from the dark recesses of the wardrobe in which it resides. She determines the surface area of material she needs to create her hat with a radius of 1 foot and a height of 0.5 feet as follows:

Anne wants to give her younger brother a Rubik's cube for his birthday, but knows that her brother has a short attention span and is easily frustrated. She custom orders a Rubik's Cube in which all the faces are black, and has to pay for the customization based on the surface area of the cube with an edge length of 4 inches.

Jeremy has a large cylindrical fish tank that he bathes in because he doesn't like showers or bathtubs. He is curious whether his heated water cools faster than when in a bathtub, and needs to calculate the surface area of his cylindrical tank of height 5.5 feet and radius of 3.5 feet.

The surface area of a capsule can be determined by combining the surface area equations for a sphere and the lateral surface area of a cylinder. Note that the surface area of the bases of the cylinder is not included since it does not comprise part of the surface area of a capsule. The total surface area is calculated as follows:

Horatio is manufacturing a placebo that purports to hone a person's individuality, critical thinking, and ability to objectively and logically approach different situations. He has already tested the market and has found that a vast majority of the sample population exhibit none of these qualities, and are very ready to purchase his product, further entrenching themselves within the traits they so desperately seek to escape. Horatio needs to determine the surface area of each capsule so that he can coat them with an excessive layer of sugar and appeal to the sugar predisposed tongues of the population in preparation for his next placebo that "cures" all forms of diabetes mellitus. Given each capsule has r of 0.05 inches and h of 0.5 inches:

The surface area of a spherical cap is based on the height of the segment in question. The calculator provided assumes a solid sphere and includes the base of the cap in the calculation of surface area, where the total surface area is the sum of the area of the base and that of the lateral surface of the spherical cap. If using this calculator to compute the surface area of a hollow sphere, subtract the surface area of the base. Given two values of height, cap radius, or base radius, the third value can be calculated using the equations provided on the Volume Calculator. The surface area equations are as follows:

Jennifer is jealous of the globe that her older brother Lawrence received for his birthday. Since Jennifer is two-thirds the age of her brother, she decides that she deserves one-third of her brother's globe. After returning her father's hand saw to the toolshed, she calculates the surface area of her hollow portion of the globe with R of 0.80 feet and h 0.53 feet as shown below:

Paul is making a volcano in the shape of a conical frustum for his science fair project. Paul views volcanic eruptions as a violent phenomenon, and being against all forms of violence, decides to make his volcano in the form of a closed conical frustum that does not erupt. Although his volcano is unlikely to impress the science fair judges, Paul must still determine the surface area of material he needs to coat the outer wall of his volcano with R of 1 foot, r of 0.3 feet, and h of 1.5 feet:

Calculating the surface area of an ellipsoid does not have a simple, exact formula such as a cube or other simpler shape does. The calculator above uses an approximate formula that assumes a nearly spherical ellipsoid:

Coltaine has always enjoyed cooking and recently won a ceramic knife from a contest. Unfortunately for his family, who almost exclusively eat meat, Coltaine has been practicing his cutting technique on an excessive amount of vegetables. Rather than eating his vegetables, Coltaine's father stares dejectedly at his plate, and estimates the surface area of the elliptical cuts of zucchini with axes 0.1, 0.2, and 0.35 inches:

The surface area of a square pyramid is comprised of the area of its square base and the area of each of its four triangular faces. Given height h and edge length a, the surface area can be calculated using the following equations:

Vonquayla's classroom recently completed building a model of the Great Pyramid of Giza. However, she feels that the model does not exude the feeling of architectural wonder that the original does and decides that coating it with "snow" would at least impart an aspect of wonder. She calculates the surface area of melted sugar she would need to fully coat the pyramid with edge length a of 3 feet and height h of 5 feet: 041b061a72




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