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In geometry, the circumference (from Latin circumferēns 'carrying around, circling') 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.^[1] 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 circumference of a sphere is the circumference, or length, of any one of its great circles.

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.^[2] 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. In modern terms, one considers a set $A$ of all perimeters of inscribed polygons, a set $B$ of all perimeters of polygons circumscribing the circle. Every element of $A$ is less than every element of $B$, and there is no positive gap: no number that is less than all differences of elements of $A$ from elements of $B$. Therefore the circumference can be defined, and calculated, as the unique number $x$ such that $a<x<b$ for every inscribed perimeter $a$ and every circumscribed perimeter $b$.

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 ellipse, $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1,$ is $C_{ellipse}\sim \pi \sqrt{2(a^2+b^2)}.$ Some lower and upper bounds on the circumference of the canonical ellipse with $a\ge b$ are:^[10] $2\pi b\le C\le 2\pi a,$ $\pi (a+b)\le C\le 4(a+b),$ $4\sqrt{a^2+b^2}\le C\le \pi \sqrt{2(a^2+b^2)}.$

Here the upper bound $2\pi a$ is the circumference of a circumscribed concentric circle passing through the endpoints of the ellipse's major axis, and the lower bound $4\sqrt{a^2+b^2}$ 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.^[11] More precisely, $C_{ellipse}=4a{\int }_0^{\pi /2}\sqrt{1-e^2{\sin }^2\theta }d\theta ,$ where $a$ is the length of the semi-major axis and $e$ is the eccentricity $\sqrt{1-b^2/a^2}.$

• Arc length – Distance along a curve
• Area – Size of a two-dimensional surface
• Circumgon – Geometric figure which circumscribes a circle
• Isoperimetric inequality – Geometric inequality applicable to any closed curve
• Perimeter-equivalent radius – Radius of a circle or sphere equivalent to a non-circular or non-spherical objectPages displaying short descriptions of redirect targets

The Wikibook Geometry has a page on the topic of: Arcs

Look up circumference in Wiktionary, the free dictionary.

• Numericana - Circumference of an ellipse