Circular segment: Difference between revisions

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In [[geometry]], a '''circle segment''' (also ''circular 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 cut-off part notbetween containingthe secant and an arc, excluding the circle's center point.

==Formula==

{{math-stub}}

[[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

<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 sector is <center><math>\pi R^2 \cdot \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]]

2 triangles with the area <center><math>\frac{1}{2} \sin \frac{\theta}{2} \cos \frac{\theta}{2}</math></center> or

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

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

Since the area of the segment is the area of the sector decreased by the area of the triangular portion, we have

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

According to trigonometry, '''<big><math>\sin x \cos x = 2\sin x\cos x=2\sin x</math></big>''', therefore

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

The area is therefore:

<math>R^2\left(\frac{\theta}{2}-\frac{1}{2}\sin\theta\right)</math>

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

==Related topics==

*[[Circular sector]]

==External links==

*[http://mathworld.wolfram.com/CircularSegment.html MathWorld's definition of "circular segment"]

[[nl:Cirkelsegment]]

[[Category:Geometry]]