Circular segment: Difference between revisions

Revision as of 12:02, 31 May 2005

Revision as of 02:01, 3 June 2005

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In [[geometry]], a '''circlecircular segment''' (also ''circular'circle segment''') 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]]. The circle segment constitutes the part between the secant and an arc, excluding the circle's center.

==FormulaFormulae==

[[Image:Circle segment.jpg|frame|right|A circular segment (shown here in yellow) is enclosed between a secant/chord (the dashed line) and the arc whose endpoints equal the chord's (the arc shown above the yellow area).]]

[[Image:Circle segment.jpg|center]]

Let '''R''' be the [[radius]] of the [[circle]], '''c''' the chord [[length]], '''s''' the arc length, '''h''' the [[height]] of the segment, and '''d''' the height of the [[triangle|triangular]] portion. The [[area]] of the circular segment is equal to the area of the [[circular sector]] minus the area of the triangular portion. From this, we know that the radius is

Let '''R''' be the [[radius]] of the [[circle]], '''c''' the chord [[length]], '''s''' the arc length, '''h''' the [[height]] of the segment, and '''d''' the height of the [[triangle|triangular]] portion. The [[area]] of the circular segment is equal to the area of the [[circular sector]] minus the area of the triangular portion. From this, we know that the radius is

<center><big>'''<math>R = s + d</math>''',</big></center>

the arc length is

<center><big>'''<math>s=R\theta</math>'''</big></center>

The area of the circular sectorradius is <center>&nbsp;<math>\pi R^2 \cdot= h + d \frac{\theta}{2\pi}</math></center>

or <center><math>R^2\left(\frac{\theta}{2}\right)</math></center>

If we bisect angle <math>\theta</math>, and thus the triangular portion, we will get

[[Image:Circle cos.jpg|center|300px]]

2The trianglesarc withlength the area <center>is&nbsp;<math>\frac{1}{2}s \sin= R \frac{\theta}{2} \cos \frac{\theta}{2}</math></center> or

orThe <center>area is&nbsp;<math>A = \frac{R^2}{2}\left(\frac{theta-\sin\theta}{2}\right)</math></center>

<br clear=all />

==Derivation of the area formula==

The area of the circular sector is&nbsp;<math>\pi R^2 \cdot \frac{\theta}{2\pi} = R^2\left(\frac{\theta}{2}\right)</math>

[[Image:Circle segmentcos.jpg|centerright|260px]]

If we bisect angle <math>\theta</math>, and thus the triangular portion, we will get two triangles with the area <math>\frac{1}{2} \sin \frac{\theta}{2} \cos \frac{\theta}{2}</math> or

<math>2\cdot\frac{1}{2}\sin\frac{\theta}{2} \cos\frac{\theta}{2}</math>

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<math>= \frac{R^2}{2}\left(\theta-\sin\theta\right)</math>

==RelatedSee topicsalso==

*[[Circular sector]]