Circumference

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

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 ${\displaystyle \pi .}$ Its first few decimal digits are 3.141592653589793... 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 equivalent to ${\displaystyle 2\pi }$. This is also the number 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 (written as ${\displaystyle C/d,}$ 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. 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.

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. If ${\displaystyle E}$ is a subset of the Euclidean plane, its one-dimensional Hausdorff measure is obtained by covering ${\displaystyle E}$ with sets of small diameter and taking the infimum of the sums of those diameters.

For sufficiently regular curves, this measure agrees with the usual notion of arc length. Equivalently, if ${\displaystyle \gamma :[a,b]\to {\mathbb{R}}^2}$ is a rectifiable curve, its length may be defined as $${\displaystyle L(\gamma )=sup\sum _{i=1}^n|\gamma (t_i)-\gamma (t_{i-1})|,}$$ where the supremum is taken over all partitions ${\displaystyle a=t_0<t_1<\cdots <t_n=b.}$ For continuously differentiable curves this length is equal to $${\displaystyle L(\gamma )={\int }_a^b|{\gamma }^{\prime }(t)|,dt.}$$

Applying this to the circle of radius (r), with parametrization $${\displaystyle \gamma (t)=(r\cos t,r\sin t),0\le t\le 2\pi ,}$$ gives ${\displaystyle |{\gamma }^{\prime }(t)|=r}$, and hence $${\displaystyle L(\gamma )={\int }_0^{2\pi }r,dt=2\pi r.}$$ 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.