Geometry
Projecting a sphere to a plane
Outline History (Timeline)
Branches Euclidean Non-Euclidean Elliptic Spherical Hyperbolic Non-Archimedean geometry Projective Affine Synthetic Analytic Algebraic Arithmetic Diophantine Differential Riemannian Symplectic Discrete differential Complex Finite Discrete/Combinatorial Digital Convex Computational Fractal Incidence Noncommutative geometry Noncommutative algebraic geometry
Concepts Features Dimension Straightedge and compass constructions Angle Curve Diagonal Orthogonality (Perpendicular) Parallel Vertex Congruence Similarity Symmetry
Zero-dimensional Point
One-dimensional Line segment ray Length
Two-dimensional Plane Area Polygon Triangle Altitude Hypotenuse Pythagorean theorem Parallelogram Square Rectangle Rhombus Rhomboid Quadrilateral Trapezoid Kite Circle Diameter Circumference Area
Three-dimensional Volume Cube cuboid Cylinder Dodecahedron Icosahedron Octahedron Pyramid Platonic Solid Sphere Tetrahedron
Four- / other-dimensional Tesseract Hypersphere
Geometers
by name Aida Aryabhata Ahmes Alhazen Apollonius Archimedes Atiyah Baudhayana Bolyai Brahmagupta Cartan Coxeter Descartes Euclid Euler Gauss Gromov Hilbert Huygens Jyeṣṭhadeva Kātyāyana Khayyám Klein Lobachevsky Manava Minkowski Minggatu Pascal Pythagoras Parameshvara Poincaré Riemann Sakabe Sijzi al-Tusi Veblen Virasena Yang Hui al-Yasamin Zhang List of geometers
by period BCE Ahmes Baudhayana Manava Pythagoras Euclid Archimedes Apollonius 1–1400s Zhang Kātyāyana Aryabhata Brahmagupta Virasena Alhazen Sijzi Khayyám al-Yasamin al-Tusi Yang Hui Parameshvara 1400s–1700s Jyeṣṭhadeva Descartes Pascal Huygens Minggatu Euler Sakabe Aida 1700s–1900s Gauss Lobachevsky Bolyai Riemann Klein Poincaré Hilbert Minkowski Cartan Veblen Coxeter Present day Atiyah Gromov
v t e

In geometry, the circumference (from Latin circumferens, meaning "carrying around") is the perimeter of a circle or ellipse.^[1] The circumference is the arc length of the circle, as if it were opened up and straightened out to a line segment.^[2] 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.

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.^[3] The term circumference is used when measuring physical objects, as well as when considering abstract geometric forms.

Relationship with π

The circumference of a circle is related to one of the most important mathematical constants. This constant, pi, is represented by the Greek letter ${\displaystyle \pi .}$ The first few decimal digits of the numerical value of ${\displaystyle \pi }$ are 3.141592653589793 ...^[4] Pi is defined as the ratio of a circle's circumference ${\displaystyle C}$ to its diameter ${\displaystyle d:}$

$${\displaystyle \pi ={\frac {C}{d)).}$$

Or, equivalently, as the ratio of the circumference to twice the radius. The above formula can be rearranged to solve for the circumference:

$${\displaystyle {C}=\pi \cdot {d}=2\pi \cdot {r}.\!}$$

The ratio of the circle's circumference to its radius is called the circle constant, and is equivalent to ${\displaystyle 2\pi }$. The value ${\displaystyle 2\pi }$ is also the amount of radians in one turn. The use of the mathematical constant π is ubiquitous in mathematics, engineering, and science.

In Measurement of a Circle written circa 250 BCE, Archimedes showed that this ratio (${\displaystyle C/d,}$ since he did not use the name π) was greater than 310/71 but less than 31/7 by calculating the perimeters of an inscribed and a circumscribed regular polygon of 96 sides.^[5] This method for approximating π 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.

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,

$${\displaystyle {\frac {x^{2)){a^{2))}+{\frac {y^{2)){b^{2))}=1,}$$

$${\displaystyle C_{\rm {ellipse))\sim \pi {\sqrt {2\left(a^{2}+b^{2}\right))).}$$

${\displaystyle a\geq b}$

$${\displaystyle 2\pi b\leq C\leq 2\pi a,}$$

$${\displaystyle \pi (a+b)\leq C\leq 4(a+b),}$$

$${\displaystyle 4{\sqrt {a^{2}+b^{2))}\leq C\leq \pi {\sqrt {2\left(a^{2}+b^{2}\right))).}$$

Here the upper bound ${\displaystyle 2\pi a}$ is the circumference of a circumscribed concentric circle passing through the endpoints of the ellipse's major axis, and the lower bound ${\displaystyle 4{\sqrt {a^{2}+b^{2))))$ 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.^[7] More precisely,

$${\displaystyle C_{\rm {ellipse))=4a\int _{0}^{\pi /2}{\sqrt {1-e^{2}\sin ^{2}\theta ))\ d\theta ,}$$

${\displaystyle a}$

${\displaystyle e}$

${\displaystyle {\sqrt {1-b^{2}/a^{2))}.}$