How would you define a 3d angle?

Question

Today I somehow was wondering how could you define a 3d angle. First what I could think of was volume of a cone with side a=1. Then if it wasn't round, volume of a tetrahedron with side 1. Is any of this fisible? How is it defined, if this field exists?

Answer 1

Well one property of the 2-dimensional angle is that the length of a circle's arc $l$ is proportional to the angle of the arc $\theta$ for a given radius $r$, i.e. if we put $r=1$ then $$l=k \theta$$ where $k$ is a proportionality constant (radians have been defined such that $k=1$ when $\theta$ is in those units, and $k=\pi/180$ if you work in degrees).

To define your 3-dimensional angle, you could choose it to have a similar property. In this case, you would have the area of a spherical cap $A$ proportional to the 3d angle of the cap $\Omega$ for a given radius, i.e. for $r=1$ $$A=\kappa \Omega$$ And such angle has a name, it is called a solid angle and the units in which $\kappa =1$ is called a steradian.

Answer 2

A 3d angle is an angle between 2 planes in space. For example a valley gutter. You essentially need to know the coordinates of 4 points, 2 of which are common for both planes.