Editing Solid angle (section)

=== Cone, spherical cap, hemisphere  ===
[[Image:Steradian cone and cap.svg|thumb|right|250px|Diagram showing a section through the centre of a cone (1) subtending a solid angle of 1 steradian in a sphere of radius r, along with the spherical "cap" (2). The external surface area A of the cap equals <math>r^2</math> only if solid angle of the cone is exactly 1 steradian. Hence, in this figure {{math|''θ'' {{=}} ''A''/2}} and {{math|''r'' {{=}} 1}}.]]
The solid angle of a [[cone (geometry)|cone]] with its apex at the apex of the solid angle, and with [[Apex (geometry)|apex]] angle 2{{math|''θ''}}, is the area of a [[spherical cap]] on a [[unit sphere]]

<math display=block>\Omega = 2\pi \left (1 - \cos\theta \right)\ = 4\pi \sin^2 \frac{\theta}{2}.</math>

For small {{math|''θ''}} such that {{math|cos ''θ'' ≈ 1 − {{sfrac|''θ''<sup>2</sup>|2}}}} this reduces to {{math|π''θ''<sup>2</sup>  ≈  π''r''<sup>2</sup>}}, the area of a circle. (As {{math|h → 0, ''θ'' → r}}.)

The above is found by computing the following [[double integral]] using the unit [[Spherical coordinate system#Integration and differentiation in spherical coordinates|surface element in spherical coordinates]]:

<math display=block>\begin{align}
\int_0^{2\pi} \int_0^\theta \sin\theta' \, d \theta' \, d \phi
&= \int_0^{2\pi} d \phi\int_0^\theta \sin\theta' \, d \theta' \\
&= 2\pi\int_0^\theta \sin\theta' \, d \theta' \\
&= 2\pi\left[ -\cos\theta' \right]_0^{\theta} \\
&= 2\pi\left(1 - \cos\theta \right).
\end{align}</math>

This formula can also be derived without the use of [[calculus]]. 

Over 2200 years ago [[Archimedes]] proved that the surface area of a spherical cap is always equal to the area of a circle whose radius equals the distance from the rim of the spherical cap to the point where the cap's axis of symmetry intersects the cap.<ref>{{cite journal |year = 2015 |title = Archimedes on Spheres and Cylinders |journal = Math Pages |url = http://www.mathpages.com/home/kmath343/kmath343.htm}}</ref> [[File:Archimedes-spherical-cap.png|thumb|Archimedes' theorem that surface area of the region of sphere below horizontal plane H in given diagram is equal to area of a circle of radius t.]]In the above coloured diagram this radius is given as
<math display=block> 2r \sin \frac{\theta}{2}. </math>
In the adjacent black & white diagram this radius is given as "t".

Hence for a unit sphere the solid angle of the spherical cap is given as

<math display=block> \Omega = 4\pi \sin^2 \frac{\theta}{2} = 2\pi \left (1 - \cos\theta \right). </math>

{{anchor|hemisphere}}
When {{math|''θ''}} = {{sfrac|{{pi}}|2}}, the spherical cap becomes a [[Sphere|hemisphere]] having a solid angle 2{{pi}}.

The solid angle of the complement of the cone is
 
<math display=block>4\pi - \Omega = 2\pi \left(1 + \cos\theta \right) = 4\pi\cos^2 \frac{\theta}{2}.</math>

This is also the solid angle of the part of the [[celestial sphere]] that an astronomical observer positioned at latitude {{math|''θ''}} can see as the Earth rotates. At the equator all of the celestial sphere is visible; at either pole, only one half.

The solid angle subtended by a segment of a spherical cap cut by a plane at angle {{mvar|''γ''}} from the cone's axis and passing through the cone's apex can be calculated by the formula<ref name = Mazonka>{{cite arXiv |last =  Mazonka |first = Oleg |year = 2012 |title = Solid Angle of Conical Surfaces, Polyhedral Cones, and Intersecting Spherical Caps |eprint=1205.1396 |class = math.MG}}</ref>

<math display=block> \Omega = 2 \left[ \arccos \left(\frac{\sin\gamma}{\sin\theta}\right) - \cos\theta \arccos\left(\frac{\tan\gamma}{\tan\theta}\right) \right]. </math>

For example, if {{math|1=''γ'' = −''θ''}}, then the formula reduces to the spherical cap formula above: the first term becomes {{pi}}, and the second {{math|{{pi}} cos ''θ''}}.