NationMaster - Encyclopedia: Steradian

The steradian (symbol: sr) is the SI unit of solid angle. It is used to describe two-dimensional angular spans in three-dimensional space, analogous to the way in which the radian describes angles in a plane. The name is derived from the Greek stereos for "solid" and the Latin radius for "ray, beam". Look up si, Si, SI in Wiktionary, the free dictionary. ... A solid angle is the three dimensional analog of the ordinary angle. ... 2-dimensional renderings (ie. ... For the musical group, see Radian (band). ... This article is about the mathematical construct. ... Latin was the language originally spoken in the region around Rome called Latium. ...

The steradian is dimensionless because 1 sr = m²·m^-2 = 1. It is useful, however, to distinguish between dimensionless quantities of different nature, so in practice the symbol "sr" is used where appropriate, rather than the derived unit "1" or no unit at all. As an example, radiant intensity can be measured in watts per steradian (W·sr^-1). In dimensional analysis, a dimensionless number (or more precisely, a number with the dimensions of 1) is a pure number without any physical units. ... In physics, intensity is a measure of the time-averaged energy flux. ...

1 Definition
2 Analogue to radians
3 SI multiples
4 See also

Definition

A single unit of steradian is defined as the solid angle subtended at the center of a sphere of radius r by a portion of the surface of the sphere having an area r². A solid angle is the three dimensional analog of the ordinary angle. ... In mathematics the term subtended usually refers to the direct relationship between an angle and its arc length. ... For other uses, see Sphere (disambiguation). ... This article is about an authentication, authorization, and accounting protocol. ... This article is about the physical quantity. ...

Section of cone (1) and spherical cap (2) inside a sphere

If this area $A,$ is equal to $r^{2},$ and it corresponds to the area of a spherical cap ( $A = 2pi rh,$ ) then the relationship $frac{h}{r}=frac{1}{2pi}$ holds. Then the solid angle of the simple cone subtending an angle θ is equal to: In geometry, a spherical cap is a portion of a sphere cut off by a plane. ...

$begin{align} theta & = arccos left( frac{r-h}{r} right) & = arccos left( 1 - frac{h}{r} right) & = arccos left( 1 - frac{1}{2pi} right) approx 0.572 ,text{rad} mbox{ or } 32.77^circ end{align}$

This angle corresponds to an apex angle of 2θ ≈ 1.144 rad or 65.54°.

Because the surface area of this sphere is 4πr², the definition implies that a sphere measures 4π steradians. By the same argument, the maximum solid angle that can be subtended at any point is 4π sr. A steradian can also be called a squared radian.

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 or 3282.80635 square degrees. Look up polygon in Wiktionary, the free dictionary. ... Angle excess is the amount by which the sum of the angles of a polygon on a sphere exceeds the sum of the angles of a polygon with the same number of sides in a plane. ... When a circles diameter is 1, its circumference is Ï€. Pi or Ï€ is the ratio of a circles circumference to its diameter in Euclidean geometry, approximately 3. ... For other uses, see Sphere (disambiguation). ... A square degree is a non-SI unit that can be used to measure solid angles (that is, the area of the projection of a surface onto a unit sphere centered on the point of observation). ...

The radian was formerly an SI supplementary unit, but this category was abolished from the SI in 1995 and the steradian is now considered an SI derived unit. Until 1995, SI (International System of Units) supplementary units were: As of October 1995, the category of supplementary units has been abolished from the SI system of measurement, and the radian and the steradian are now considered SI derived units. ... Look up si, Si, SI in Wiktionary, the free dictionary. ... SI derived units are part of the SI system of measurement units and are derived from the seven SI base units. ...

Analogue to radians

In two dimensions, the angle in radians is related to the arc length it cuts out: Determining the length of an irregular arc segmentâ€”also called rectification of a curveâ€”was historically difficult. ...

$theta = frac{s}{r} ,$

where

s is arc length, and

r is the radius of the circle.

Now in three dimensions, the solid angle in steradians is related to the area it cuts out:

$Omega = frac{S}{r^2} ,$

where

S is the surface area, and

r is the radius of the sphere.

SI multiples

Multiple	Name	Symbol
10⁰	steradian	sr
10^–1	decisteradian	dsr
10^–2	centisteradian	csr
10^–3	millisteradian	msr
10^–6	microsteradian	µsr
10^–9	nanosteradian	nsr
10^–12	picosteradian	psr
10^–15	femtosteradian	fsr
10^–18	attosteradian	asr
10^–21	zeptosteradian	zsr
10^–24	yoctosteradian	ysr

Contents

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Analogue to radians

SI multiples

See also

Definition