Area
Area is the measure of the size of a surface, or of a region on such a surface. In analytical geometry, the letter "A" commonly denotes formulas for the determination of the areas of simple two-dimensional plane figures, while the letter "S" denotes formulas for the determination of surface areas of three-dimensional solids. The concept of area is dependent upon the more fundamental quantity of length. In ancient civilizations, people measured various distances by determining the number of times a standard unit of length must be added together to yield the distance in question. For example, the ancient Romans used the "stadium", with about 8.7 stadia equal to one statute mile, as their "reference" length to measure large distances. About 1,045 stadia (or 120 statute miles) added together equals the straight-line distance between the two Italian cities of Rome and Naples. In the same way, a surface is measured by denoting the number of times a standard unit of area must be added together to equal the area in question. For instance, the playing area of a football field is equal to 48,000 square feet (300 feet of length times 160 feet of width), or about 5,333.33 square yards. As implied in this example, the standard geometric shape for measuring area is the square. Since all four sides of a square are the same length, the length of one side uniquely identifies the size of the square. To determine the area of a rectangle, the lengths of its width and height are measured and multiplied together. The resulting quantity represents the number of identical squares which, when added together, equals the area of the rectangle. For instance, if a rectangle's width is nine centimeters (cm) and its height is 4 cm, then the rectangle's area is "9 cm x 4 cm" (36 cm2), or is equal to the combined area of 36 identical squares, each square having sides of 1 cm.
For other simple plane figures (like triangles) and the surface areas of simple solids (like spheres), formulas have been derived to calculate the area of each using their linear dimensions. For instance, the area of a right triangle can be calculated with the formula "1/2ab" where "a" represents the length of the triangle's base and "b" represents the length of the triangle's altitude. If a triangle's base equals nine centimeters and its height equals six centimeters, its area is "1/2 x 9 cm x 6 cm," or 27 cm2.
As an especially important example of area determination, the area of a regular polygon (a figure of three or more straight sides of equal length joined at equal angles to one another) is defined as the sum of the areas of triangles into which it can be decomposed. Around the third century b.c., Archimedes of Syracuse, who established the determination of areas (then called the "method of exhaustion" by its inventor Eudoxus of Cnidus in the fourth century b.c.), utilized the method of inscribing equilateral polygons inside a circle. Then, upon increasing the number of their sides, the areas of the polygons (which Archimedes could calculate) approached the area of the circle as a limit. Using this result together with a similar idea involving circumscribed polygons, Archimedes was able to find the area of the circle as "A = r2," in which "r" is the radius of the circle and "pi" is a constant that Archimedes calculated to possess a value between 3-1/7 and 3-10/71. Then, in 1635 Bonaventura Cavalieri, a professor of mathematics at the University of Bologna, formulated a systematic method for the determination of areas. His method involved complicated geometric considerations. Later, Isaac Barrow, the Lucasian professor of mathematics at Cambridge, published in 1670 a treatise that helped to unify formulations of areas within the mathematical branches of analytic geometry and integral calculus. Barrow used the traditional approach of geometry to calculate areas and was thus prevented from taking the final step to calculate areas with the use of integral calculus, as was later developed by Sir Isaac Newton of England and Gottfried Wilheim Leibniz of Germany.
The areas of irregular figures, both of planar figures and the surfaces of solids, can be computed by the use of calculus. In two dimensions, this mathematical tool involves the computation of the area bounded between a curve having the equation "y = ƒ(x)," the horizontal x-axis, and the distance between the vertical lines "x=a" and "x=b". This type of computation is commonly called the limit between the points "a" and "b", where a<b. The area of an irregularly shaped figure can be found by subdividing it into rectangles of equal width. If the number of rectangles is made larger and larger (therefore, their bases [or widths] become smaller and smaller), the sum of their areas (found by multiplying base by height) approaches the required area as a limit. By using this general method, integral calculus provides a systematic way for obtaining an exact calculation of many areas of irregular figures. The notation used in integral calculus to indicate total area of a region by summing the areas of these discrete rectangles is of the form: ƒ(x) dx. This expression calculates the surface area of the region bounded by the continuous curve described by the function ƒ(x) and the intervals "x = a" to "x = b" (where a<b), and the x-axis. Where the integral of a function (of the form just described) cannot be "analytically" (i.e., explicitly) solved, numerical approximations can be found using recursive (or algorithmic) methods. The digital computer has, of course, found great application to such problems.
The system of measurement most commonly used by scientists and engineers throughout the world is the System International (SI), commonly referred to as the "metric system". Reference units of length, mass, charge, etc. are precisely defined within this system. The standard unit of length is the meter, from which the basic unit of area in the metric system is derived; namely, the square meter (denoted m2). The square meter is defined as the area of a square whose sides are one meter long. Larger or smaller units of area can be formulated from the square meter. Two examples are the square kilometer, 1 km2, which equals (or is composed of) 106 m2 and the square centimeter, 1 cm2, which equals 10-4 m2.
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