In geometry, the circumference (from Latin circumferentia, meaning "carrying around") of a circle is the (linear) distance around it.[1] That is, the circumference would be the length of the circle if it were opened up and straightened out to a line segment. Since a circle is the edge (boundary) of a disk, circumference is a special case of perimeter.[2] The perimeter is the length around any closed figure and is the term used for most figures excepting the circle and some circularlike figures such as ellipses. Informally, "circumference" may also refer to the edge itself rather than to the length of the edge. Contents 1
Circumference
1.1 Relationship with π 2
Circumference
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 can not 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. When a circle's diameter is 1, its circumference is π. When a circle's radius is 1—called a unit circle—its circumference is 2π. Relationship with π[edit] The circumference of a circle is related to one of the most important mathematical constants. This constant, pi, is represented by the Greek letter π. The first few decimal digits of the numerical value of π are 3.141592653589793 ...[4] Pi is defined as the ratio of a circle's circumference C to its diameter d: π = C 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: C = π ⋅ d = 2 π ⋅ r . displaystyle C =pi cdot d =2pi cdot r .! The use of the mathematical constant π is ubiquitous in mathematics,
engineering, and science.
In Measurement of a
Circle
x 2 a 2 + y 2 b 2 = 1 , displaystyle frac x^ 2 a^ 2 + frac y^ 2 b^ 2 =1, is C e l l i p s e ∼ π 2 ( a 2 + b 2 ) . displaystyle C_ rm ellipse sim pi sqrt 2(a^ 2 +b^ 2 ) . Some lower and upper bounds on the circumference of the canonical ellipse with a ≥ b displaystyle ageq b are[6] 2 π b ≤ C ≤ 2 π a , displaystyle 2pi bleq Cleq 2pi a, π ( a + b ) ≤ C ≤ 4 ( a + b ) , displaystyle pi (a+b)leq Cleq 4(a+b), 4 a 2 + b 2 ≤ C ≤ π 2 ( a 2 + b 2 ) . displaystyle 4 sqrt a^ 2 +b^ 2 leq Cleq pi sqrt 2(a^ 2 +b^ 2 ) . Here the upper bound 2 π a displaystyle 2pi a is the circumference of a circumscribed concentric circle passing through the endpoints of the ellipse's major axis, and the lower bound 4 a 2 + b 2 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, we have C e l l i p s e = 4 a ∫ 0 π / 2 1 − e 2 sin 2 θ d θ , displaystyle C_ rm ellipse =4aint _ 0 ^ pi /2 sqrt 1e^ 2 sin ^ 2 theta dtheta , where again a displaystyle a is the length of the semimajor axis and e displaystyle e is the eccentricity 1 − b 2 / a 2 . displaystyle sqrt 1b^ 2 /a^ 2 .
Circumference
Arc length Area Isoperimetric inequality References[edit] ^
San Diego State University
External links[edit] The Wikibook
Geometry
Look up circumference in Wiktionary, the free dictionary. Numericana 
Circumference
