In
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, the circumference (from Latin ''circumferens'', meaning "carrying around") is the
perimeter of a
circle or
ellipse. That is, the circumference would be the
arc length
ARC may refer to:
Business
* Aircraft Radio Corporation, a major avionics manufacturer from the 1920s to the '50s
* Airlines Reporting Corporation, an airline-owned company that provides ticket distribution, reporting, and settlement services
* ...
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
Limit or Limits may refer to:
Arts and media
* ''Limit'' (manga), a manga by Keiko Suenobu
* ''Limit'' (film), a South Korean film
* Limit (music), a way to characterize harmony
* "Limit" (song), a 2016 single by Luna Sea
* "Limits", a 2019 ...
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.
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 The first few decimal digits of the numerical value of
are 3.141592653589793 ... Pi is defined as the
ratio of a circle's circumference
to its
diameter
In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints lie on the circle. It can also be defined as the longest chord of the circle. Both definitions are also valid f ...
Or, equivalently, as the ratio of the circumference to twice the
radius. The above formula can be rearranged to solve for the circumference:
The use of the mathematical constant is ubiquitous in mathematics, engineering, and science.
In ''
Measurement of a Circle'' written circa 250 BCE,
Archimedes showed that this ratio (
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
Christoph (Christophorus) Grienberger (also variously spelled Gruemberger, Bamberga, Bamberger, Banbergiera, Gamberger, Ghambergier, Granberger, Panberger) (2 July 1561 – 11 March 1636) was an Austrian Jesuit astronomer, after whom the crater ...
who used polygons with 10
40 sides.
Ellipse
Circumference is used by some authors 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 ellipse,
is
Some lower and upper bounds on the circumference of the canonical ellipse with
are:
Here the upper bound
is the circumference of a
circumscribed concentric circle passing through the endpoints of the ellipse's major axis, and the lower bound
is the
perimeter of an
inscribed
{{unreferenced, date=August 2012
An inscribed triangle of a circle
In geometry, an inscribed planar shape or solid is one that is enclosed by and "fits snugly" inside another geometric shape or solid. To say that "figure F is inscribed in figu ...
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
In integral calculus, an elliptic integral is one of a number of related functions defined as the value of certain integrals, which were first studied by Giulio Fagnano and Leonhard Euler (). Their name originates from their originally arising i ...
.
More precisely,
where
is the length of the semi-major axis and
is the eccentricity
See also
*
*
*
*
*
References
External links
Numericana - Circumference of an ellipse
{{Authority control
Geometric measurement
Circles