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Quantum Statistics 50 : Solid Angle Steradian
Quantum Statistics 50 : Solid Angle Steradian
Published: 2013/08/16
Channel: Adam Beatty
steradian as unit of solid angle in urdu animated
steradian as unit of solid angle in urdu animated
Published: 2017/07/31
Channel: Jan visual physics
Plane angle and Solid angle
Plane angle and Solid angle
Published: 2016/06/08
Channel: Makrani Education
steradian as unit of solid angle animated
steradian as unit of solid angle animated
Published: 2017/08/01
Channel: Jan visual physics
Steradian
Steradian
Published: 2016/01/22
Channel: WikiAudio
SOLID ANGLE
SOLID ANGLE
Published: 2017/04/28
Channel: 7activestudio
Calculus 3: Triple Integrals (16 of 25) What is a Steradian?
Calculus 3: Triple Integrals (16 of 25) What is a Steradian?
Published: 2017/06/02
Channel: Michel van Biezen
Steradians, Radians Squared, Degrees Squares as Solid Angles
Steradians, Radians Squared, Degrees Squares as Solid Angles
Published: 2014/11/17
Channel: Andrew Ochadlick
Steradian
Steradian
Published: 2014/10/11
Channel: Audiopedia
Radiant Flux, Radiance and Solid Angle
Radiant Flux, Radiance and Solid Angle
Published: 2012/09/24
Channel: UCDavis
1. Physics | Solid Angle | Concept of Solid Angle | by Ashish Arora
1. Physics | Solid Angle | Concept of Solid Angle | by Ashish Arora
Published: 2016/02/25
Channel: Physics Galaxy
LUMEN
LUMEN
Published: 2017/03/27
Channel: 7activestudio
Steradian Gallery 0.1.2
Steradian Gallery 0.1.2
Published: 2017/06/11
Channel: kromond
Steradian Semi Imaging Radar
Steradian Semi Imaging Radar
Published: 2017/10/24
Channel: Steradian Semiconductors
Topic 20 Part 8   Steradians
Topic 20 Part 8 Steradians
Published: 2015/11/17
Channel: ECET 364 - Fundamentals of Electromagnetics
What does steradian mean?
What does steradian mean?
Published: 2015/08/17
Channel: What Does That Mean?
Steradian Meaning
Steradian Meaning
Published: 2015/04/13
Channel: SDictionary
Introduction to Radians (3 of 3: Definition + Why Radians Aren
Introduction to Radians (3 of 3: Definition + Why Radians Aren't Units)
Published: 2015/11/07
Channel: Eddie Woo
Steradian EM System Software 4D Radar Point Cloud Extraction for a traffic scene
Steradian EM System Software 4D Radar Point Cloud Extraction for a traffic scene
Published: 2017/04/19
Channel: Steradian Semiconductors
Fun with Steradian Laser Tag!
Fun with Steradian Laser Tag!
Published: 2016/09/21
Channel: Lafayette Macaroni Kid
Laser Tag Birthday Party with Steradian Laser Tag
Laser Tag Birthday Party with Steradian Laser Tag
Published: 2012/12/13
Channel: SteradianLaserTag
How to Pronounce Steradian
How to Pronounce Steradian
Published: 2015/06/02
Channel: Pronunciation Guide
Steradian 3d angle..... By_Manoj pathak MAU
Steradian 3d angle..... By_Manoj pathak MAU
Published: 2016/05/02
Channel: Saurabh Rai
Supplementary units in Urdu  ::1st year Chapter 1 Physics in Urdu
Supplementary units in Urdu ::1st year Chapter 1 Physics in Urdu
Published: 2016/03/30
Channel: SWAP Education Portal
Steradian Lasertag Machining S-7
Steradian Lasertag Machining S-7
Published: 2012/10/10
Channel: SteradianLaserTag
What are Radians? (Old Video)
What are Radians? (Old Video)
Published: 2014/12/20
Channel: Don't Memorise
Woods Laser Tag by Steradian
Woods Laser Tag by Steradian
Published: 2012/12/12
Channel: SteradianLaserTag
Steradian "Watts Per Steradian" Live at The Sunset
Steradian "Watts Per Steradian" Live at The Sunset
Published: 2011/07/26
Channel: LanthanideSeries
Kama Sutra and Cryptography - Talk at IIT by M. N. Krish, Author of The Steradian Trail
Kama Sutra and Cryptography - Talk at IIT by M. N. Krish, Author of The Steradian Trail
Published: 2014/05/03
Channel: M. N. Krish
Solid angle explanation with derivation
Solid angle explanation with derivation
Published: 2015/04/26
Channel: Ion academy
What is a radian?
What is a radian?
Published: 2011/06/08
Channel: Jonathan Mitchell
Concept of Solid Angle| Class 12| IITJEE PHYSICS VIDEO LECTURES |B.M SHARMA
Concept of Solid Angle| Class 12| IITJEE PHYSICS VIDEO LECTURES |B.M SHARMA
Published: 2016/03/06
Channel: IIT JEE PHYSICS MEDICAL VIDEO LECTURES
radian as unit of plane angle in urdu animated
radian as unit of plane angle in urdu animated
Published: 2017/07/30
Channel: Jan visual physics
Steradian EM System Software 4D Radar Point Cloud Extraction for a revolving human
Steradian EM System Software 4D Radar Point Cloud Extraction for a revolving human
Published: 2017/04/19
Channel: Steradian Semiconductors
Equinox Rx Series: Yoga Tune Up Trigger Point Therapy
Equinox Rx Series: Yoga Tune Up Trigger Point Therapy
Published: 2013/06/03
Channel: Furthermore from Equinox
HexBolt Sphere with Steradians
HexBolt Sphere with Steradians
Published: 2016/04/19
Channel: Louis Coffelt
Steradian Top  #5 Facts
Steradian Top #5 Facts
Published: 2016/02/02
Channel: Rishika Janaki
Concept of Solid angle (B.Sc)
Concept of Solid angle (B.Sc)
Published: 2017/08/25
Channel: Guru Kpo
Παρουσίαση Εικονικής Περιήγησης - Steradians 360
Παρουσίαση Εικονικής Περιήγησης - Steradians 360
Published: 2015/04/29
Channel: Μύρων Γουρλής
Steradian Laser Tag Referee Education
Steradian Laser Tag Referee Education
Published: 2016/06/21
Channel: SteradianLaserTag
Steradian Reflex Barriers
Steradian Reflex Barriers
Published: 2012/02/22
Channel: SteradianLaserTag
How to say "steradian"! (High Quality Voices)
How to say "steradian"! (High Quality Voices)
Published: 2017/02/11
Channel: WordBox
How to Pronounce Steradian
How to Pronounce Steradian
Published: 2014/01/13
Channel: Emma Saying
Calculus 3: Triple Integrals (17 of 25) Calculate the Angle of a Steradian
Calculus 3: Triple Integrals (17 of 25) Calculate the Angle of a Steradian
Published: 2017/06/03
Channel: Michel van Biezen
Steradian Laser Tag Disc Skeet
Steradian Laser Tag Disc Skeet
Published: 2012/12/11
Channel: SteradianLaserTag
Steradian, Inc.
Steradian, Inc.
Published: 2010/10/18
Channel: Yuzaki Shin
Steradian - Seattle, May 2012
Steradian - Seattle, May 2012
Published: 2012/05/27
Channel: Glenn Smith
Steradian - Light Speed
Steradian - Light Speed
Published: 2011/04/27
Channel: LanthanideSeries
How To Pronounce Steradian
How To Pronounce Steradian
Published: 2017/10/19
Channel: Emma Saying
Richard Bullen - Steradian (2014) for 12 Trombones
Richard Bullen - Steradian (2014) for 12 Trombones
Published: 2014/08/05
Channel: BULLEN MUSIC
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WIKIPEDIA ARTICLE

From Wikipedia, the free encyclopedia
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Steradian
Steradian.svg
A graphical representation of 1 steradian.
The sphere has radius r, and in this case the area A of the highlighted surface patch is r2. The solid angle Ω equals [A/r2] sr which is 1 sr in this example. The entire sphere has a solid angle of 4πsr.
Unit information
Unit system SI derived unit
Unit of Solid angle
Symbol ㏛ 

The steradian (symbol: sr) or square radian[1][2] is the SI unit of solid angle. It is used in three-dimensional geometry, and is analogous to the radian which quantifies planar angles. The name is derived from the Greek stereos for "solid" and the Latin radius for "ray, beam".

The steradian, like the radian, is a dimensionless unit, essentially because a solid angle is the ratio between the area subtended and the square of its distance from the vertex: both the numerator and denominator of this ratio have dimension length squared (i.e. L2/L2 = 1, dimensionless). It is useful, however, to distinguish between dimensionless quantities of a different nature, so the symbol "sr" is used to indicate a solid angle. For example, radiant intensity can be measured in watts per steradian (W·sr−1). The steradian was formerly an SI supplementary unit, but this category was abolished in 1995 and the steradian is now considered an SI derived unit.

Definition[edit]

Section of cone (1) and spherical cap (2) that subtend a solid angle of one steradian inside a sphere

A steradian can be defined as the solid angle subtended at the center of a unit sphere by a unit area on its surface. For a general sphere of radius r, any portion of its surface with area A = r2 subtends one steradian.[3]

The solid angle is related to the area it cuts out of a sphere:

where
A is the surface area of the spherical cap, ,
r is the radius of the sphere, and
sr is the unit, steradian.

Because the surface area A of a sphere is 4πr2, the definition implies that a sphere measures 4π (≈ 12.56637) steradians. By the same argument, the maximum solid angle that can be subtended at any point is 4π sr.

Other properties[edit]

Since A = r2, it corresponds to the area of a spherical cap (A = 2πrh) (wherein h stands for the "height" of the cap), and the relationship h/r = 1/2π holds. Therefore one steradian corresponds to the plane (i.e. radian) angle of the cross-section of a simple cone subtending the plane angle 2θ, with θ given by:

This angle corresponds to the plane aperture angle of 2θ ≈ 1.144 rad or 65.54°.

A steradian is also equal to the spherical area of a polygon having an angle excess of 1 radian, to 1/4π of a complete sphere, or to (180/π)2 ≈ 3282.80635 square degrees.

The solid angle of a cone whose cross-section subtends the angle 2θ is:

.

Analogy to radians[edit]

In two dimensions, an angle is related to the length of the circular arc that it spans:

where
l is arc length,
r is the radius of the circle, and
rad is the radian.

Similarly in three dimensions, the solid angle is related to the area of the spherical surface that it spans:

where
A is the surface area of the spherical cap, 2πrh,
r is the radius of the sphere, and
sr is the steradian.

For example, a measurement of the width of a visible object would be given in radians. At the same time its visible area over one's visible field would be given in steradians. Just as the area of a circle is quadratically related to its diameter or radius, so is the area of the piece of a spherical surface cut out by a circular cone related to the angle across the piece. That area, or equivalently the proportion of a sphere its cone takes up, is a measure of three dimensional proportional volume, analogous to the two dimensional measure over the circle represented by an angle.

One-dimensional circular measure has units of radians or degrees, while two-dimensional spherical measure is expressed in steradians. In higher dimensional mathematical spaces, units for analogous solid angles have not been explicitly named. When they are used, they are dealt with by analogy with the circular or spherical cases. That is, as a proportion of the relevant unit hypersphere taken up by the generalized angle, or point set expressed in spherical coordinates.

SI multiples[edit]

A complete sphere subtends 4π ≈ 12.56637 steradians, so multiples larger than the decasteradian are rarely used.

Any range in excess of the whole area of a sphere would only be needed in conjunction with non-Euclidean, spherical geometry. Then it would also usually only be required in the context of nonlocal analysis. As such, solid angles over 4π are used exceedingly rarely in the literature.

See also[edit]

Notes[edit]

References[edit]

  1. ^ Stutzman, Warren L; Thiele, Gary A (2012-05-22). Antenna Theory and Design. ISBN 978-0-470-57664-9. 
  2. ^ Woolard, Edgar (2012-12-02). Spherical Astronomy. ISBN 978-0-323-14912-9. 
  3. ^ "Steradian", McGraw-Hill Dictionary of Scientific and Technical Terms, fifth edition, Sybil P. Parker, editor in chief. McGraw-Hill, 1997. ISBN 0-07-052433-5.

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