Steradian
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The steradian (symbol: sr) or square radian is the unit of solid angle in the
International System of Units The International System of Units, internationally known by the abbreviation SI (from French ), is the modern form of the metric system and the world's most widely used system of measurement. It is the only system of measurement with official s ...
(SI). It is used in three-dimensional geometry, and is analogous to the
radian The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. It is defined such that one radian is the angle subtended at ...
, which quantifies planar angles. A solid angle in the form of a circular cone can be projected onto a sphere from its centre, delineating a spherical cap where the cone intersects the sphere. The magnitude of the solid angle expressed in steradians is defined as the quotient of the surface area of the spherical cap and the square of the sphere's radius. This is analogous to the way a plane angle projected onto a circle delineates a circular arc on the circumference, whose length is proportional to the angle. Steradians can be used to measure a solid angle of any projected shape. The solid angle subtended is the same as that of a cone with the same projected area. A solid angle of one steradian subtends a cone aperture of approximately 1.144 radians or 65.54 degrees. In the SI, solid angle is considered to be a dimensionless quantity, the ratio of the area projected onto a surrounding sphere and the square of the sphere's radius. This is the number of square radians in the solid angle. This means that the SI steradian is the number of square radians in a solid angle equal to one square radian, which of course is the number one. It is useful to distinguish between dimensionless quantities of a different kind, such as the radian (in the SI, a ratio of quantities of dimension length), so the symbol sr is used. 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. The name ''steradian'' is derived from the Greek 'solid' + radian.


Definition

A steradian can be defined as the solid angle subtended at the centre of a unit sphere by a unit area (of any shape) on its surface. For a general sphere of
radius In classical geometry, a radius (: radii or radiuses) of a circle or sphere is any of the line segments from its Centre (geometry), center to its perimeter, and in more modern usage, it is also their length. The radius of a regular polygon is th ...
, any portion of its surface with area subtends one steradian at its centre. A solid angle in the form of a circular cone is related to the area it cuts out of a sphere: : \Omega = \frac\ \text \, = \frac\ \text, where * is the solid angle * is the surface area of the spherical cap, , * is the radius of the sphere, * is the height of the cap, and * sr is the unit, steradian; sr = rad. Because the surface area of a sphere is , the definition implies that a sphere subtends steradians (≈ 12.56637 sr) at its centre, or that a steradian subtends  ≈ 0.07958 of a sphere. By the same argument, the maximum solid angle that can be subtended at any point is  sr.


Other properties

The area of a spherical cap is , where is the "height" of the cap. If , then . From this, one can compute the cone aperture (a plane angle) of the cross-section of a simple spherical cone whose solid angle equals one steradian: : \theta = \arccos \left( \frac \right) = \arccos \left( 1 - \frac \right) = \arccos \left( 1 - \frac \right) , giving ≈ 0.572 rad 32.77° and aperture ≈ 1.144 rad 65.54°. The solid angle of a spherical cone whose cross-section subtends the angle is: : \Omega = 2\pi(1 - \cos\theta) \text = 4\pi\sin^2\left(\frac\right) \text. A steradian is also equal to of a complete
sphere A sphere (from Ancient Greek, Greek , ) is a surface (mathematics), surface analogous to the circle, a curve. In solid geometry, a sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...
( spat), to ≈ 3282.80635  square degrees, and to the spherical area of a polygon having an angle excess of 1 radian.


SI multiples

Millisteradians (msr) and microsteradians (μsr) are occasionally used to describe
light Light, visible light, or visible radiation is electromagnetic radiation that can be visual perception, perceived by the human eye. Visible light spans the visible spectrum and is usually defined as having wavelengths in the range of 400– ...
and particle beams.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) Other multiples are rarely used.


See also

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


References


External links

* {{classical mechanics derived SI units Natural units SI derived units Units of solid angle