Square degree

Physical unit
Unit name	Square degree
Unit symbol	${\displaystyle \mathrm {deg^{2}} }$, , ${\displaystyle \mathrm {sq.deg.} }$ ${\displaystyle \mathrm {(^{\circ })^{2}} }$
Physical quantity (s)	Solid angle
Formula symbol	${\displaystyle \Omega }$
dimension	${\displaystyle {\mathsf {{\frac {L^{2}}{L^{2}}}=1}}}$
In SI units	${\displaystyle \mathrm {1\ deg^{2}={\frac {\pi ^{2}}{32\,400}}\;sr} }$

Square degrees ( deg² , (°) ² ) ( English square degree or sq degree ) is a (non-legal) unit for the solid angle . It is mainly used in astronomy as a measure of the extent of an object in the sky .

The sub-units are called accordingly:

• Square arc minute ( English square arcmin )  square degrees ${\displaystyle ={\frac {1}{3600}}}$
• Square arcsecond ( English square arcsec )  square minute  square degrees. ${\displaystyle ={\frac {1}{3600}}}$ ${\displaystyle ={\frac {1}{12\,960\,000}}}$

The full solid angle of the celestial sphere is sr = ${\displaystyle 4\pi }$ ${\displaystyle {\frac {(360\ \mathrm {deg} )^{2}}{\pi }}\approx 41\,253\ \mathrm {deg^{2}} }$

The size of a celestial object in square degrees results from the proportion of the celestial sphere that the object covers, multiplied by 41253.

The conversion of a square degree into steradian results in:

${\displaystyle {\begin{aligned}\Leftrightarrow 1\ \mathrm {deg} ^{2}&=\left({\frac {2\pi }{360}}\right)^{2}\ \mathrm {sr} \\&=\left({\frac {\pi }{180}}\right)^{2}\ \mathrm {sr} \\&={\frac {\pi ^{2}}{32\,400}}\ \mathrm {sr} \\&\approx 3{,}0461741\cdot 10^{-4}\ \mathrm {sr} \\&\approx 0{,}00030461741\ \mathrm {sr} \end{aligned}}}$

or.

${\displaystyle \Leftrightarrow 1\ \mathrm {sr} \approx 3283\ \mathrm {deg} ^{2}}$

For small angles , solid angles can be calculated approximately as with surfaces: an object whose extension measures 1 degree in one direction and 1 degree perpendicular to it, therefore covers a solid angle of around 1 square degree.

• 

  Solid angle as a function of the opening angle of a cone

• 

  enlarged section of the function for opening angles 0–45 °

See also

• Facial field