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Difference between revisions of "Segment"

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A segment on a plane is a plane figure included between a curve and a chord of it. The area of a segment of a circle (a circular segment) is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083790/s0837901.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083790/s0837902.png" /> is the radius of the circle and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083790/s0837903.png" /> is the length of the arc.
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A segment on a plane is a plane figure included between a curve and a chord of it. The area of a segment of a circle (a circular segment) is $S=r^2(\theta-\sin\theta)/2$, where $r$ is the radius of the circle and $r\theta$ is the length of the arc.
  
A segment in space is a part of a solid bounded by a plane and the part of the surface cut off by the plane. The volume <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083790/s0837904.png" /> of a segment of a ball (a spherical segment) is given by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083790/s0837905.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083790/s0837906.png" /> is the radius of the ball and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083790/s0837907.png" /> is the height of the segment. The area <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083790/s0837908.png" /> of the curved surface of a segment of a ball is given by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083790/s0837909.png" />.
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A segment in space is a part of a solid bounded by a plane and the part of the surface cut off by the plane. The volume $V$ of a segment of a ball (a spherical segment) is given by $V=\pi h^2(3R-h)/3$, where $R$ is the radius of the ball and $h$ is the height of the segment. The area $S$ of the curved surface of a segment of a ball is given by $S=2\pi Rh$.
  
  

Latest revision as of 13:34, 9 April 2014

A segment on a plane is a plane figure included between a curve and a chord of it. The area of a segment of a circle (a circular segment) is $S=r^2(\theta-\sin\theta)/2$, where $r$ is the radius of the circle and $r\theta$ is the length of the arc.

A segment in space is a part of a solid bounded by a plane and the part of the surface cut off by the plane. The volume $V$ of a segment of a ball (a spherical segment) is given by $V=\pi h^2(3R-h)/3$, where $R$ is the radius of the ball and $h$ is the height of the segment. The area $S$ of the curved surface of a segment of a ball is given by $S=2\pi Rh$.


Comments

Of course, a line segment is the part of a line between two of its points, or (in real projective geometry) one of the two parts into which two points decompose the line through them [a1], pp. 176-177.

For a segment in space see also [a2], p.245.

References

[a1] H.S.M. Coxeter, "Introduction to geometry" , Wiley (1989)
[a2] H. Lamb, "Infinitesimal calculus" , Cambridge (1924)
How to Cite This Entry:
Segment. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Segment&oldid=12070
This article was adapted from an original article by A.B. Ivanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article