Steradian

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## Definition

## Other properties

## SI multiples

## See also

## Notes

## References

## External links

This article uses material from the Wikipedia page available here. It is released under the Creative Commons Attribution-Share-Alike License 3.0.

Steradian

Steradian | |
---|---|

General information | |

Unit system | SI derived unit |

Unit of | Solid angle |

Symbol | sr |

Conversions | |

SI base units | 1 m^{2}/m^{2} |

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. Whereas an angle in radians, projected onto a circle, gives a *length* on the circumference, a solid angle in steradians, projected onto a sphere, gives an *area* on the surface. The name is derived from the Greek ? *stereos* 'solid' + radian.

The steradian, like the radian, is a dimensionless unit, the quotient of the area subtended and the square of its distance from the center. Both the numerator and denominator of this ratio have dimension length squared (i.e. , 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.

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 subtends one steradian at its center.^{[3]}

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

- where
- ? is the solid angle
*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 4*?r*^{2}, the definition implies that a sphere subtends 4*?* steradians (? 12.56637 sr) at its center. By the same argument, the maximum solid angle that can be subtended at any point is 4*?* sr.

If *A* = *r*^{2}, it corresponds to the area of a spherical cap (*A* = 2*?rh*) (where *h* stands for the "height" of the cap) and the relationship *h*/*r* = 1/2*?* holds. Therefore, in this case, one steradian corresponds to the plane (i.e. radian) angle of the cross-section of a simple cone subtending the plane angle 2*?*, with *?* given by:

This angle corresponds to the plane aperture angle of 2*?* ? 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 1/4*?* of a complete sphere, or to (180°/*?*)^{2}_{} ? 3282.80635 square degrees.

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

- .

Millisteradians (msr) and microsteradians (?sr) are occasionally used to describe light and particle beams.^{[4]}^{[5]} Other multiples are rarely used.

**^**Stutzman, Warren L; Thiele, Gary A (2012-05-22).*Antenna Theory and Design*. ISBN 978-0-470-57664-9.**^**Woolard, Edgar (2012-12-02).*Spherical Astronomy*. ISBN 978-0-323-14912-9.**^**"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.**^**Stephen M. Shafroth, James Christopher Austin,*Accelerator-based Atomic Physics: Techniques and Applications*, 1997, ISBN 1563964848, p. 333**^**R. Bracewell, Govind Swarup, "The Stanford microwave spectroheliograph antenna, a microsteradian pencil beam interferometer"*IRE Transactions on Antennas and Propagation***9**:1:22-30 (1961)

- Media related to Steradian at Wikimedia Commons

This article uses material from the Wikipedia page available here. It is released under the Creative Commons Attribution-Share-Alike License 3.0.

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