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== {{anchor|Square minute|Square second}}Definition and properties ==
{{See also|Spherical polygon area}}
The magnitude of an object's solid angle in [[steradian]]s is equal to the [[area (geometry)|area]] of the segment of a [[unit sphere]], centered at the apex, that the object covers. Giving the area of a segment of a unit sphere in steradians is analogous to giving the length of an arc of a [[unit circle]] in radians. Just as the magnitude of a plane angle in radians at the vertex of a circular sector is the ratio of the length of its arc to its radius, the magnitude of a solid angle in steradians is the ratio of the area covered on a sphere by an object to the square of the radius of the sphere. The formula for the magnitude of the solid angle in steradians is 

<math display=block>\Omega=\frac{A}{r^2}, </math>

where <math>A</math> is the area (of any shape) on the surface of the sphere and <math>r</math> is the radius of the sphere.

Solid angles are often used in [[astronomy]], [[physics]], and in particular [[astrophysics]]. The solid angle of an object that is very far away is roughly proportional to the ratio of area to squared distance. Here "area" means the area of the object when projected along the viewing direction.

[[Image:Solid_Angle,_1_Steradian.svg|thumb|Any area on a sphere which is equal in area to the square of its radius, when observed from its center, subtends precisely one [[steradian]].]]

The solid angle of a sphere measured from any point in its interior is 4[[Pi|{{pi}}]]&nbsp;sr. The solid angle subtended at the center of a cube by one of its faces is one-sixth of that, or 2{{pi}}/3&nbsp; sr. {{anchor|Octant}}The solid angle subtended at the corner of a cube (an [[octant (geometry)|octant]]) or spanned by a [[spherical octant]] is {{pi}}/2&nbsp; sr, one-eighth of the solid angle of a sphere.

Solid angles can also be measured in [[square degree]]s (1&nbsp;sr = {{pars|180/{{pi}}}}<sup>2</sup> square degrees), in square [[arc-minutes]] and square [[arc-seconds]].{{efn|The whole sphere contains ~148.510 million square arcminutes and ~534.638 billion square arcseconds.{{Citation needed|date=June 2025}}}} It can also be expressed in fractions of the sphere (1&nbsp;sr = {{sfrac|1|4{{pi}}}} fractional area), also known as [[spat (angular unit)|spat]] (1&nbsp;sp = 4{{pi}}&nbsp;sr).

In [[spherical coordinates#Integration_and_differentiation_in_spherical_coordinates|spherical coordinates]] there is a formula for the [[Differential of a function|differential]],

<math display=block>d\Omega = \sin\theta\,d\theta\,d\varphi,</math>

where {{mvar|θ}} is the [[colatitude]] (angle from the North Pole) and {{mvar|φ}} is the longitude.

The solid angle for an arbitrary [[oriented surface]] {{mvar|S}} subtended at a point {{mvar|P}} is equal to the solid angle of the projection of the surface {{mvar|S}} to the unit sphere with center {{mvar|P}}, which can be calculated as the [[surface integral]]:

<math display=block>\Omega = \iint_S \frac{ \hat{r} \cdot \hat{n}}{r^2}\,dS \ = \iint_S \sin\theta\,d\theta\,d\varphi,</math>

where <math>\hat{r} = \vec{r} / r</math> is the [[unit vector]] corresponding to <math> \vec{r} </math>, the [[position vector]] of an infinitesimal area of surface {{math|''dS''}} with respect to point {{mvar|P}}, and where <math> \hat{n} </math> represents the unit [[normal vector]] to {{math|''dS''}}. Even if the projection on the unit sphere to the surface {{mvar|S}} is not [[isomorphic]], the multiple folds are correctly considered according to the surface orientation described by the sign of the [[scalar product]] <math>\hat{r} \cdot \hat{n}</math>.

Thus one can approximate the solid angle subtended by a small [[facet]] having flat surface area {{math|''dS''}}, orientation <math>\hat{n}</math>, and distance {{math|''r''}} from the viewer as:

<math display=block>d\Omega = 4 \pi \left(\frac{dS}{A}\right) \, (\hat{r} \cdot \hat{n}),</math>

where the [[surface area of a sphere]] is {{math|1=''A'' = 4{{pi}}''r''<sup>2</sup>}}.