Circumference
The general definition of circumference is a line or external boundary of a closed curvilinear figure or object. The more common definition of circumference within mathematics is the measure of the outer boundary (commonly called the perimeter) of an elliptical area, especially a circular area. The letter "C" commonly denotes the length of a circle's circumference. The circumference of a sphere is defined as the length of any great circle on the sphere. A great circle is the circle on the surface of a sphere produced by a plane that passes through the center of the sphere.
One of the earliest references to circumference was by Heraclitus of Ephesos (535-475 b.c.) when he used the word "periphereia" to mean "The beginning and end join on the circumference of the circle." Circumference most likely is a combination of the Latin "circum" (around) and "ferre" (to carry) and a derivation of "circumferentia" (to carry around) that is a Latin translation of the earlier Greek term "periphereia."
A very simple way to directly measure the approximate circumference of a circle is to wrap a string around the figure, pull the string out straight, and then measure the string's length.
A historical approach to finding an approximation to the circumference of a circle involves an iterative calculation of the perimeters of regular polygons inscribed in the circle. Given a circle, let pn be the perimeter of a regular polygon of n sides inscribed in the circle. Then as n gets larger and larger, the number pn increases. (That is, the perimeter of each term in the sequence pn is greater than the preceding term.) From calculus, if a sequence of numbers is increasing (that is, if each term in the sequence is greater than the preceding term), and if the sequence is bounded (that is, if there is a number that is equal to or greater than any term in the sequence), then the sequence has a limit. With the pn sequence possessing those attributes, the circumference of a circle, therefore, is the limit of the sequence of perimeters pn of the inscribed regular polygons, that is, C = pn.
To calculate the circumference of a circle several common terms need to be defined. The point equidistant from all the points on the circumference of a circle is called the center of the circle. The straight-line segment (or interval) from the center of the circle to a point on the circumference is called a radius. A diameter is a straight line through the center of the circle with its two ends on the circumference and of length twice that of a radius of the circle. With regards to the radius, r, or diameter, d, of a circle, the equation to solve for circumference, C, is C = 2r = d, where pi is a constant number that is defined as the ratio of a circle's circumference to its diameter. As an example, if the radius of a circle is measured to be 20 centimeters (cm) and the approximate value of 3.14 is used for pi, then the circumference is calculated to be "2 x 3.14 x 20 cm", or C = 125.6 cm.
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