Contact between peoples with different cultures, usually leading to change in one or both systems. Forms of culture contact traditionally include acculturation, assimilation, and amalgamation. Acculturation is the process of change in material culture, traditional practices, and beliefs that occurs when one group interferes in the cultural system of another, directly or indirectly challenging the latter to adapt to the ways of the former. Such change has characterized most political conquests and expansions over the centuries. Assimilation is the process whereby individuals or groups of differing ethnicity are absorbed into the dominant culture of a society—though not always completely. In the U.S. millions of European immigrants became assimilated within two or three generations; factors included the upheaval of overseas relocation, the influences of the public school system, and other forces in American life. Amalgamation (or hybridization) occurs when a society becomes ethnically mixed in a way that represents a synthesis rather than the elimination or absorption of one group by another. In Mexico, for example, Spanish and Indian cultures became increasingly amalgamated over centuries of contact.
Learn more about culture contact with a free trial on Britannica.com.
|Square||is the length of the side of the square.|
|Regular triangle||is the length of one side of the triangle.|
|Regular hexagon||is the length of one side of the hexagon.|
|Regular octagon||is the length of one side of the octagon.|
|Any regular polygon||is the apothem, or the radius of an inscribed circle in the polygon, and is the perimeter of the polygon.|
|Any regular polygon||is the Perimeter and is the number of sides.|
|Any regular polygon (using degree measure)||is the Perimeter and is the number of sides.|
|Rectangle||and are the lengths of the rectangle's sides (length and width).|
|Parallelogram (in general)||and are the length of the base and the length of the perpendicular height, respectively.|
|Rhombus||and are the lengths of the two diagonals of the rhombus.|
|Triangle||and are the base and altitude (measured perpendicular to the base), respectively.|
|Triangle||and are any two sides, and is the angle between them.|
|Circle||or||is the radius and the diameter.|
|Ellipse||and are the semi-major and semi-minor axes, respectively.|
|Trapezoid||and are the parallel sides and the distance (height) between the parallels.|
|Total surface area of a Cylinder||and are the radius and height, respectively.|
|Lateral surface area of a cylinder||and are the radius and height, respectively.|
|Total surface area of a Cone||and are the radius and slant height, respectively.|
|Lateral surface area of a cone||and are the radius and slant height, respectively.|
|Total surface area of a Sphere||or||and are the radius and diameter, respectively.|
|Total surface area of an ellipsoid||See the article.|
|Circular sector||and are the radius and angle (in radians), respectively.|
|Square to circular area conversion||is the area of the square in square units.|
|Circular to square area conversion||is the area of the circle in circular units.|
The area of irregular polygons can be calculated using the "Surveyor's formula".
It remains to show that the notion of area thus defined does not depend on the way one subdivides a polygon into smaller parts.
A typical way to introduce area is through the more advanced notion of Lebesgue measure. In the presence of the axiom of choice it is possible to prove the existence of shapes whose Lebesgue measure cannot be meaningfully defined. Such 'shapes' (they cannot a fortiori be simply visualised) enter into Tarski's circle-squaring problem (and, moving to three dimensions, in the Banach–Tarski paradox). The sets involved do not arise in practical matters.
The general formula for the surface area of the graph of a continuously differentiable function where and is a region in the xy-plane with the smooth boundary: