In geometry, the circumference () is the perimeter of a circle or ellipse. The circumference is the arc length of the circle, as if it were opened up and straightened out to a line segment. More generally, the perimeter is the curve length around any closed figure. Circumference may also refer to the circle itself, that is, the locus corresponding to the edge of a disk. The is the circumference, or length, of any one of its great circles.

Circle

The circumference of a circle is the distance around it, but if, as in many elementary treatments, distance is defined in terms of straight lines, this cannot be used as a definition. Under these circumstances, the circumference of a circle may be defined as the limit of the perimeters of inscribed regular polygons as the number of sides increases without bound. The term circumference is used when measuring physical objects, as well as when considering abstract geometric forms.

Archimedes described how the circumference can be defined and computed using the method of exhaustion of all perimeters of inscribed polygons. Therefore the circumference can be defined, and calculated, as the unique number.

Relationship with

The circumference of a circle is related to one of the most important mathematical constants. This constant, pi, is represented by the Greek letter \pi. Its first few decimal digits are 3.141592653589793... Pi is defined as the ratio to its diameter

Or, equivalently, as the ratio of the circumference to twice the radius

. This is also the number of radians in one turn. The use of the mathematical constant is ubiquitous in mathematics, engineering, and science.

In Measurement of a Circle written circa 250 BCE, Archimedes since he did not use the name ) was greater than 3 but less than 3 by calculating the perimeters of an inscribed and a circumscribed regular polygon of 96 sides. This method for approximating was used for centuries, obtaining more accuracy by using polygons of larger and larger number of sides. The last such calculation was performed in 1630 by Christoph Grienberger who used polygons with 1040 sides.

Circumference in analysis

In modern mathematical analysis, the circumference of a circle can be defined as its one-dimensional measure. One common intrinsic definition is by Hausdorff measure with sets of small diameter and taking the infimum of the sums of those diameters.

is a rectifiable curve

Thus, in analysis, the formula for circumference follows from a general definition of one-dimensional length rather than from assuming in advance that the circle has a perimeter given by polygonal exhaustion.

Ellipse

Some authors use circumference to denote the perimeter of an ellipse. There is no general formula for the circumference of an ellipse in terms of the semi-major and semi-minor axes of the ellipse that uses only elementary functions. However, there are approximate formulas in terms of these parameters. One such approximation, due to Euler (1773), for the canonical are:

is the circumference of a circumscribed concentric circle is the perimeter of an inscribed rhombus with vertices at the endpoints of the major and minor axes.

The circumference of an ellipse can be expressed exactly in terms of the complete elliptic integral of the second kind.

See also

Notes

References

External links

• Numericana - Circumference of an ellipse