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What is the conversion between degrees and steradians? That is, if you rotate a two-dimensional angle around one side, what is the equivalent in solid angle?
I specifically need the number of degrees in radians that, when rotated about the axis, form a cone with a solid angle of one steradian:
The "cone," of course, has a spherical end cap.
The solid angle subtended by an angle $\alpha$ at the center of the unit sphere is
$$2 \pi \int_0^{\alpha} d\theta \, \sin{\theta} = 2 \pi (1-\cos{\alpha})$$
When this is $1$ str, then
$$\alpha = \arccos{\left(1-\frac{1}{2 \pi}\right)} \approx 0.572 \,\text{rad}$$
or about $32.8^{\circ}$.
