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".

The steradian, like the radian, 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).

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.

If this area, A, is equal to r2 and it corresponds to the area of a spherical cap (A = 2πrh,) then the relationship $frac\left\{h\right\}\left\{r\right\}=frac\left\{1\right\}\left\{2pi\right\}$ holds. Then the solid angle of the simple cone subtending an angle θ is equal to:

:

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 4πsr. 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.

The steradian 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.

In two dimensions, the angle in radians is related to the arc length it cuts out:::$heta = frac\left\{s\right\}\left\{r\right\} ,$: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:::$Omega = frac\left\{S\right\}\left\{r^2\right\} ,$:where::"S" is the surface area, and::"r" is the radius of the sphere.

I multiples

*Solid angle

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• stéradian — [ steradjɑ̃ ] n. m. • 1923; angl. steradian (1881); du gr. stereos « solide » (cf. stéréo ) et radian ♦ Unité de mesure d angle solide (symb.sr). ● stéradian nom masculin ( …   Encyclopédie Universelle

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• steradian — [stə rā′dē ən] n. [ STE(REO) + RADIAN] the basic unit of solid angular measure in the SI system, equal to the solid angle subtended at the center of a sphere by an area, on its surface, that is equal to the square of its radius: one sphere equals …   English World dictionary

• Stéradian — Le stéradian (symbole : sr) est l unité dérivée du système international pour la mesure d angles solides. Sa définition est similaire à celle du radian qui mesure les angles plans. Son nom est partiellement dérivé du grec ancien στερεός… …   Wikipédia en Français

• Steradian — steradianas statusas T sritis Standartizacija ir metrologija apibrėžtis Erdvinio kampo išvestinio SI matavimo vieneto specialus pavadinimas. Steradianas – tai erdvinis kampas su viršūne sferos centre, išpjaunantis sferos paviršiuje figūrą,… …   Penkiakalbis aiškinamasis metrologijos terminų žodynas

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• steradian — (sr)    the standard unit of solid angle measure in mathematics. Just as there are 2pi radians in a circle, there are 4pi steradians in a sphere. Thus one steradian equals about 0.079 577 sphere. There are 129 600/pi = 41 252.96 square degrees in …   Dictionary of units of measurement

• steradian — /steuh ray dee euhn/, n. Geom. a solid angle at the center of a sphere subtending a section on the surface equal in area to the square of the radius of the sphere. Abbr.: sr [1880 85; STE(REO) + RADIAN] * * * ▪ unit of measurement       unit of… …   Universalium