We all know how to construct at least we have idea on how to construct the angle of 1 radian,but for spatial angle 1 steradian, which is defined as the angle that departs from the Centers sphere and in the sphere covers an area as the square of its radius.It would be interesting if anyone has any sketch looks like the angle of one steradian, probably resembles any cone or any other form.
- Anybody can ask a question
- Anybody can answer
- The best answers are voted up and rise to the top
We're used to associating angles with a certain shape because in the plane all angles that have the same measure are congruent to each other. This is not the case with steradians. A steradian is a measure of a spatial angle and doesn't correspond to any particular shape; as you say, it could be the aperture of a cone or of any other form.
A particularly simple form to visualize that one might actually construct in practice relies on the interesting fact that the surface area of a sphere is precisely equal to that of its projection onto a cylinder containing the sphere and touching it. That is, for instance, if you cut a sphere with two planes parallel to the $x$-$y$ plane, the surface cut out of the sphere has the same area as the surface cut out of a cylinder with its axis in the $z$ direction passing through the centre of the sphere.
That surface area is easily calculated. For a unit sphere, the radius of the cylinder is also $1$, so its perimeter is $2\pi$, and a slab of cylindrical surface of height $h$ has surface area $2\pi h$. Thus, a slab of height $1/(2\pi)$ has surface area $1$, so a chunk of sphere between two parallel planes at a distance $1/(2\pi)$ also has surface area $1$, and therefore subtends a spatial angle of $1$ steradian.