# Steradian

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. 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 στερεός *stereos* 'solid' + radian.

steradian | |
---|---|

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} |

The steradian, like the radian, 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 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

A steradian can be defined as the solid angle subtended at the centre of a unit sphere by a circular 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 1/4π (≈ 0.07958) of a sphere. By the same argument, the maximum solid angle that can be subtended at any point is 4*π* sr.

## Other properties

If *A* = *r*^{2}, it corresponds to the area of a spherical cap (*A* = 2*πrh*) (where *h* stands for the "height" of the cap) and the relationship *h*/*r* = 1/2*π* 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*θ*, with *θ* given by:

This angle corresponds to the plane aperture angle of 2*θ* ≈ 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*π* 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 2*θ* is:

## SI multiples

Millisteradians (msr) and microsteradians (μsr) are occasionally used to describe light and particle beams.[4][5] Other multiples are rarely used.

## See also

*n*-sphere- Spat (angular unit)
- IAU designated constellations by area

## References

- 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)

## External links

*in Wiktionary, the free dictionary.*

**steradian**- Media related to Steradian at Wikimedia Commons