= Solid angle of a circular cone
{title2=$\Omega=2\pi(1-\cos\theta)$}
A <right circular cone> with vertex $O$ selects all directions within a half-opening <angle> $\theta$ of its axis. Here $\theta$ is measured from the axis to a boundary direction, so the full opening in a meridian plane is $2\theta$. For $0\leq\theta\leq\pi$, interpret the selected directions as a <spherical cap>; for $\theta>\pi/2$, this is a region larger than a <hemisphere>.
The <spherical coordinate system> gives the <area element> $dA=R^2\sin\alpha\,d\alpha\,d\phi$ on the <sphere>. Dividing by $R^2$ and integrating over the cap yields
$$
d\Omega=\sin\alpha\,d\alpha\,d\phi,\qquad
\Omega=\int_0^{2\pi}\int_0^\theta\sin\alpha\,d\alpha\,d\phi
=2\pi(1-\cos\theta).
$$
The factor $\sin\alpha$ matters: a thin angular strip near a pole contains fewer directions than an equally wide strip near the equator. Treating equal intervals of both <spherical coordinates> as equal <solid angles> would overcount the polar directions.
\Image[/solid-angle-cone-and-spherical-cap.png]
{title=One set of directions cuts similar <spherical caps> from <spheres> of different <radii>}
{description=Original geometric construction with half-opening <angle> $45^\circ$. Left: a meridian section of two concentric <spheres>; rotating each blue arc about the dotted axis gives its cap. Right: the corresponding patch on the unit <sphere>. The patch is curved, and its <surface area>, rather than the volume inside the cone, determines the <solid angle>.}
At $\theta=45^\circ$, the <solid angle> is $\pi(2-\sqrt2)\approx1.8403$ <steradians>, or about $14.64\%$ of all directions. At $\theta=90^\circ$ it is $2\pi$ <steradians>, and at $180^\circ$ it is $4\pi$ <steradians>. For a small $\theta$ measured in <radians>, the <Taylor series> $\cos\theta=1-\theta^2/2+O(\theta^4)$ gives the <small-angle approximation> $\Omega\approx\pi\theta^2$. Doubling a narrow half-opening <angle> therefore approximately quadruples its coverage.
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