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The **steradian** (symbol: **sr**) or **square radian**^{[1]}^{[2]} is the unit of solid angle in the International System of Units (SI). It is used in three dimensional geometry, and is analogous to the radian, which quantifies planar angles. A solid angle in steradians, projected onto a sphere, gives the *area* of a spherical cap on the surface, whereas an angle in radians, projected onto a circle, gives a *length* of a circular arc on the circumference. The name is derived from the Greek στερεός *stereos* 'solid' + radian.

steradian | |
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General information | |

Unit system | SI |

Unit of | solid angle |

Symbol | sr |

Conversions | |

1 sr in ... | ... is equal to ... |

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

square degrees | 180^{2}/π^{2} deg^{2}≈ 3282.8 deg ^{2} |

The steradian is a dimensionless unit, the quotient of the area subtended and the square of its distance from the centre. Both the numerator and denominator of this ratio have dimension length squared (i.e. L^{2}/L^{2} = 1, dimensionless). It is useful, however, to distinguish between dimensionless quantities of a different kind, such as the radian (a ratio of quantities of dimension length), 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 centre 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* = *r*^{2} subtends one steradian at its centre.^{[3]}

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

where

- Ω is the solid angle
- A is the surface area of the spherical cap, ,
- r is the radius of the sphere,
- h is the height of the cap, and
- sr is the unit, steradian.

Because the surface area *A* of a sphere is 4*πr*^{2}, the definition implies that a sphere subtends 4*π* steradians (≈ 12.56637 sr) at its centre, or that a steradian subtends 1/4*π* ≈ 0.07958 of a sphere. By the same argument, the maximum solid angle that can be subtended at any point is 4*π* sr.

The area of a spherical cap is *A* = 2*πrh*, where h is the "height" of the cap. If *A* = *r*^{2}, then . From this, one can compute the plane aperture angle 2*θ* of the cross-section of a simple cone whose solid angle equals one steradian:

giving *θ* ≈ 0.572 rad or 32.77° and 2*θ* ≈ 1.144 rad or 65.54°.

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

A steradian is also equal to of a complete sphere (spat), to ≈ 3282.80635 square degrees, and to the spherical area of a polygon having an angle excess of 1 radian.^{[clarification needed]}

Millisteradians (msr) and microsteradians (μsr) 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)

Look up **steradian** in Wiktionary, the free dictionary.

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