# Steradian

Steradian | |
---|---|

A graphical representation of 1 steradian.The sphere has radius r, and in this case the area A of the highlighted surface patch is r^{2}. The solid angle Ω equals [A/r^{2}] 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 | sr |

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 στερεός *stereos* 'solid' + radian.

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 center: 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 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* = *r*^{2} subtends one steradian at its center.^{[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 4π*r*^{2}, the definition implies that a sphere subtends 4π steradians (≈ 12.56637 sr) at its center. By the same argument, the maximum solid angle that can be subtended at any point is 4π sr.

## Other properties

Since *A* = *r*^{2}, it corresponds to the area of a spherical cap (*A* = 2π*rh*) (where *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 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 1/4π 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 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.

Solid angles over 4π steradians—the solid angle of a full Euclidean sphere—are rarely encountered.

## See also

Look up in Wiktionary, the free dictionary.steradian |

## Notes

## 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)