Steradian

The steradian (symbol: sr) or square radian

Definition

A steradian can be defined as the solid angle subtended at the centre 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 = r² subtends one steradian at its centre.^[3]

The solid angle is related to the area it cuts out of a sphere:

$${\displaystyle \Omega ={\frac {A}{r^{2}}}\ {\text{sr}}\,={\frac {2\pi h}{r}}\ {\text{sr}},}$$

• Ω is the solid angle
• A is the surface area of the spherical cap, ${\displaystyle 2\pi rh}$,
• 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 4πr², the definition implies that a sphere subtends 4π 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 4π sr.

Other properties

If A = r², it corresponds to the area of a spherical cap (A = 2πrh, where h is the "height" of the cap) and the relationship ${\displaystyle {\tfrac {h}{r}}={\tfrac {1}{2\pi }}}$ 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 2θ, with θ given by:

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

This angle corresponds to the plane aperture angle of 2θ ≈ 1.144 rad or 65.54°.

A steradian is also equal to the spherical area of a

${\displaystyle {\tfrac {1}{4\pi }}}$

${\displaystyle \left({\tfrac {360^{\circ }}{2\pi }}\right)^{2}\approx }$

The solid angle of a cone whose cross-section subtends the angle 2θ is:

$${\displaystyle \Omega =2\pi \left(1-\cos \theta \right){\text{ sr}}=4\pi \sin ^{2}\left({\frac {\theta }{2}}\right){\text{ sr}}.}$$

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

External links

Look up steradian in Wiktionary, the free dictionary.

• Media related to Steradian at Wikimedia Commons