Caculating Area: Polygons
Calculating the area of polygons involves determining the size of the space enclosed within their boundaries, which are formed by straight line segments. Polygons are classified based on the number of sides they have, with familiar examples including triangles, rectangles, and octagons. The method for calculating the area can vary depending on the type of polygon. For instance, the area of a rectangle is found by multiplying its length by its width, while the area of a triangle is calculated as half of its base multiplied by its height. For more complex or irregular polygons, techniques such as the "Surveyor's" method—also known as Gauss's formula or the shoelace formula—are employed to divide the polygon into manageable triangles, making it easier to calculate the area. An alternative approach involves creating hypothetical triangles to transform the irregular shape into a regular polygon, then adjusting the area based on these additions. Regular polygons can be calculated more straightforwardly using the formula A = 1/2 (pa), where p is the perimeter of the inscribed circle and a is the apothem. The Bolyai-Gerwien theorem further indicates that two polygons with equal area can be rearranged into each other through cutting and reassembling, highlighting an interesting aspect of geometric properties.
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Subject Terms
Caculating Area: Polygons
Polygons are two-dimensional plane figures bound by straight line segments (the polygon's sides or edges, meeting at its corners, or vertices) in a closed circuit. Familiar polygons include triangles, rectangles, and octagons, all of which have names derived from the number of their sides or angles; this naming pattern is generalized as n-gons, where n is the polygon's number of sides, and the number of sides is the primary way in which polygons are classified. While polygons are often pictured as equilateral and/or equiangular—such as the hexagon used for stop signs—this is only one type of polygon. In mathematics, "polygon" usually refers to a subtype of polygons, the simple polygon, whose line segments do not intersect. Star polygons (sometimes called complex, self-intersecting, or coptic polygons), whose boundaries do intersect themselves, are a special and rarely considered case.
Every polygon has as many vertices as it has sides. Simple polygons may also be described as either convex, in which any line drawn through the polygon meets the boundary exactly twice, or concave. Simple polygons that are both equilateral and equiangular are called regular polygons. Regular polygons are also vertex-transitive, meaning that all vertices of the polygon are within the same symmetry orbit. Polyhedra (singular polyhedron) are the three-dimensional equivalent of polygons.
Overview
The area of a polygon is the region enclosed by its boundaries. Various means can be used to measure the area of a polygon. Some, for special cases of polygons, are quite simple and have been known for centuries. The area of a rectangle (a four-sided polygon with four right angles), for instance, is its length times its width, whereas the area of a triangle is one half of its base times its height. Means that will work for any simple polygon use a variety of techniques. The "Surveyor's" method, for instance, divides the polygon into triangles and calculates the area of each. This method is sometimes called Gauss's formula, for Carl Fredrich Gauss, or the shoelace formula, and is used in surveying, forestry, and other real-world applications, because of its efficiency in calculating the area of irregular polygons. Instead of adding up triangles, it is also possible to calculate area by finding "missing" triangles, by which method hypothetical triangles are added to the polygon in order to turn it into a regular polygon with an area that is easily calculated; the area of the missing triangles is then subtracted from that of the regular polygon.
The simplest formula for calculating the area of a regular polygon is A = 1/2 (pa), where p is the perimeter of the polygon's inscribed circle and a is its apothem, a line drawn from the center of the polygon to the midpoint of one of its sides (which is also the radius of the inscribed circle).
According to the Bolyai-Gerwien theorem, given any two polygons of equal area, one polygon can be cut into polygonal pieces and reassembled to form the second polygon.
Bibliography
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