Diameter
A diameter is a straight-line segment that passes through the center of a circle or sphere and whose two end points lie on the circumference of the surface. The center is a point of symmetry within a circle or sphere, with any diameter passing through that center denoted as an axis of symmetry. An infinite number of diameters are possible for any given circle or sphere, and are the longest line segments that terminate on the circumference. With one revolution around a circle or sphere equal to 360 degrees, the two end-points of a diameter are 180 degrees away from one another.
Diameter is an appropriate name for the measure across a circle or sphere, as the word is the union of the Greek roots "dia" (across) and "metros" (to measure). One of the earliest uses for the word diameter came from Euclid of Alexandria, a fourth-to-third century b.c. Greek mathematician, who used the word with relationship to the line bisecting a circle and also to mean the diagonal of a square.
If r is the radius of a circle or a sphere, then the diameter, d, equals d = 2r. For example, if the radius of a circle has a length of ten centimeters (cm), then the circle's diameter is "d = 2 x 10 cm" or "d = 20 cm". The diameter is also defined as d = C / , where C equals the circle's circumference and pi, , is approximately valued at 3.14. In this example, if the circumference of the circle equals a length of thirty centimeters, then the circle's diameter is "d = 30 cm / 3.14", or the diameter is approximately equal to 9.55 cm. An interesting take on the formula "d = C / " is that if C is given as, say, an integer, then that circle's diameter, d, must be irrational (it cannot be expressed by the ratio of two integers) and transcendental (it satisfies no algebraic equation with integer coefficients) because pi also possesses those properties.
The history of diameter is rooted in the attempts by the ancient Greeks to find an exact ratio for a value of pi, that of circumference over diameter (C / d). The quest of the Greeks took the form of an effort to "square the circle"; that is, to construct with straight edge and compass both a square and a circle of identical areas. A very early approximation for pi was the ancient Greek value of "22 / 7". Greek mathematician, physicist, and inventor Archimedes of Syracuse, in the third century b.c., is recorded to have made the first scientific effort to compute pi. Archimedes computed pi to a value between 3-10/71 and 3-1/7, or about 3.14. Four centuries later, at around AD 200, pi was valued to 3.1416. By the early sixth century Chinese and Indian mathematicians had independently confirmed or improved on the number of decimal places that pi was calculated past the decimal point. By the end of the seventeenth century in Europe, new methods of mathematical analysis provided various ways of calculating pi. In 1767 Johann Heinrich Lambert showed that the ratio C/d is irrational, and in 1882 Carl Louis Friedrich von Lindemann proved that an exact value for pi is impossible. Early in the twentieth century the Indian mathematician Srinivasa Ramanujan developed ways to calculate pi so efficiently that they were eventually incorporated into computer algorithms, permitting expressions of pi at, presently, billions of digits. Throughout the many centuries of gaining more precise values of pi, diameter played an important role in this development.
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