Circumference: Difference between revisions
No edit summary |
add ellipse approximation |
||
| (6 intermediate revisions by 4 users not shown) | |||
| Line 1: | Line 1: | ||
{{stub}} | {{stub}} | ||
'''Circumference''' | '''Circumference''' is essentially a synonym for [[perimeter]]: for a given [[closed curve]] in the [[plane]], it is the distance one travels in a complete circuit of the curve. The term circumference is most frequently used to refer to the distance around a [[circle]], though it may refer to the distance around any [[smooth]] curve, while the term perimeter is typically reserved for [[polygon]]s and other non curving shapes. | ||
== | ==Formulas== | ||
In a circle of [[radius]] <math>r</math> and [[diameter]] <math>d = 2r</math>, the circumference <math>C</math> is given by | |||
<cmath>C = \pi \cdot d = 2\pi \cdot r</cmath> | |||
Indeed, the [[constant]] <math>\pi</math> ([[pi]]) was originally defined to be the [[ratio]] of the circumference of a circle to the length of its diameter. | |||
There is, however, no algebraic formula for the circumference of an ellipse (without integrals). Several approximations exist, such as this one:<cmath> C \approx \pi \left(a + b\right) \left( 1 + \frac{3h}{10 + \sqrt{4 - 3h}} \right) \quad\text{where } h = \frac{\left(a - b\right)^2}{\left(a + b \right)^2}</cmath>by Ramanujan. | |||
==See Also== | |||
Latest revision as of 20:04, 3 July 2024
This article is a stub. Help us out by expanding it.
Circumference is essentially a synonym for perimeter: for a given closed curve in the plane, it is the distance one travels in a complete circuit of the curve. The term circumference is most frequently used to refer to the distance around a circle, though it may refer to the distance around any smooth curve, while the term perimeter is typically reserved for polygons and other non curving shapes.
Formulas
In a circle of radius 
and diameter 
, the circumference 
is given by
![\[C = \pi \cdot d = 2\pi \cdot r\]](http://latex.artofproblemsolving.com/3/5/8/358134d6a09f1e059e89656f96ac8faf3bb4d11a.png)
Indeed, the constant 
(pi) was originally defined to be the ratio of the circumference of a circle to the length of its diameter.
There is, however, no algebraic formula for the circumference of an ellipse (without integrals). Several approximations exist, such as this one:![\[C \approx \pi \left(a + b\right) \left( 1 + \frac{3h}{10 + \sqrt{4 - 3h}} \right) \quad\text{where } h = \frac{\left(a - b\right)^2}{\left(a + b \right)^2}\]](http://latex.artofproblemsolving.com/4/a/b/4ab217396d45db82c06976813a0cfecddcbdb4cc.png)
by Ramanujan.
