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In [[geometry]], a '''circular segment''' (symbol: '''⌓''') is an area of a [[circle]] informally defined as an area which is "cut off" from the rest of the circle by a [[secant line|secant]] or a [[chord (geometry)|chord]].
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== Formulas ==
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[[Image:Circularsegment.svg|frame|right|A circular segment (in green) is enclosed between a secant/chord (the dashed line) and the arc whose endpoints equal the chord's (the arc shown above the green area).]]
 
Let '''R''' be the [[radius]] of the [[circle]], '''θ''' is the central angle in radians, '''α''' is the central angle in degrees, '''c''' the chord [[length]], '''s''' the arc length, '''h''' the [[height]] of the segment, and '''d''' the height of the [[triangle|triangular]] portion.
 
*The radius is&nbsp;<math>R = h + d = h/2+c^2/8h \frac{}{}</math>
*The arc length is&nbsp;<math>s = \frac{\alpha}{180}\pi R = {\theta} R</math>
 
*The chord length is&nbsp;<math>c = 2R\sin\frac{\theta}{2} = R\sqrt{2-2\cos\theta}</math>
 
*The height is&nbsp;<math>h = R(1-\cos\frac{\theta}{2}) = R - \sqrt{R^2 - \frac{c^2}{4}} </math>
 
*The angle is&nbsp;<math> \theta = 2\arccos\frac{d}{R} = 2\arcsin\frac{c}{2R}</math>
 
=== Area ===
 
The [[area]] of the circular segment is equal to the area of the [[circular sector]] minus the area of the triangular portion.
 
Area of Segment = ½ × (θ - sin θ) × ''R''²  (when angle is in radians)
 
Area of Segment = ½ × ( (α × π/180) - sin α) × ''R''²  (when angle is in degrees)
 
== See also ==
* [[Circular sector]]
* [[Spherical segment]]
* [[Arc (geometry)|Arc]]
* [[Conic section]]
* [[Cross section (geometry)|Cross section]]
 
== External links ==
* {{MathWorld |urlname=CircularSegment |title=Circular segment}}
* [http://www.mathopenref.com/segment.html Definition of a circular segment] With interactive animation
* [http://www.mathopenref.com/segmentarea.html Formulae for area of a circular segment] With interactive animation
[[Category:Circles]]

Latest revision as of 04:59, 6 November 2014

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