C4.5 algorithm: Difference between revisions

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[[Image:Circle arc.svg|thumb|300px|A circular sector is shaded in green]]
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A '''circular sector''' or '''circle sector''' (symbol: '''⌔'''), is the portion of a [[disk (mathematics)|disk]] enclosed by two [[radius|radii]] and an [[Arc (geometry)|arc]], where the smaller [[area (geometry)|area]] is known as the minor sector and the larger being the major sector. In the diagram, θ is the [[central angle]] in [[radian]]s, <math>r</math> the radius of the circle, and <math>L</math> is the arc length of the minor sector.
 
A sector with the central angle of 180° is called a [[semicircle]]. Sectors with other central angles are sometimes given special names, these include '''quadrants''' (90°), '''sextants''' (60°) and '''octants''' (45°).
 
The angle formed by connecting the endpoints of the arc to any point on the circumference that is not in the sector is equal to half the central angle.
 
== Area ==
The total area of a circle is <math>\pi r^2</math>.  The area of the sector can be obtained by multiplying the circle's area by the ratio of the angle and <math>2 \pi</math> (because the area of the sector is proportional to the angle, and <math>2 \pi</math> is the angle for the whole circle):
 
:<math>A =
\pi r^2 \cdot \frac{\theta}{2 \pi} =
\frac{r^2 \theta}{2}
</math>
The area of a sector in terms of <math>L</math> can be obtained by multiplying the total area <math>\pi r^2</math>by the ratio of <math>L</math> to the total perimeter <math>2\pi r</math>.
:<math>A =
\pi r^2 \cdot \frac{L}{2\pi r}  = \frac{r \cdot L}{2}
</math>
 
Another approach is to consider this area as the result of the following integral :
:<math>A =
\int_0^\theta\int_0^r dS=\int_0^\theta\int_0^r \tilde{r} d\tilde{r} d\tilde{\theta} = \int_0^\theta \frac{1}{2} r^2 d\tilde{\theta} = \frac{r^2 \theta}{2}
</math>
 
Converting the central angle into [[degree (angle)|degree]]s gives
:<math>A = \pi r^2 \cdot \frac{\theta ^{\circ}}{360}</math>
 
== Perimeter ==
The length of the [[perimeter]] of a sector is the sum of the arc length and the two radii:
:<math>P
= L + 2r
= \theta r + 2r
= r \left( \theta + 2 \right)</math>
 
where ''&theta;'' is in radians.
 
== See also ==
*[[Circular segment]] - the part of the sector which remains after removing the triangle formed by the center of the circle and the two endpoints of the circular arc on the boundary.
*[[Conic section]]
*[[Hyperbolic sector]]
 
== References ==
*Gerard, L. J. V. ''The Elements of Geometry, in Eight Books; or, First Step in Applied Logic'', London, Longman's Green, Reader & Dyer, 1874. [http://books.google.com/books?id=4jMDAAAAQAAJ&pg=PA285&dq=%22circular+sector%22+octant&ei=BfjBS667MISUzgTK24HMAg&cd=10#v=onepage&q=%22circular%20sector%22%20octant&f=false p. 285]
 
== External links ==
* [http://www.mathopenref.com/arcsector.html Definition and properties of a circle sector] with interactive animation
* {{MathWorld |urlname=CircularSector |title=Circular sector}}
 
[[Category:Circles]]

Latest revision as of 03:31, 20 December 2014

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