What is a 3d angle ?

A 3D angle, also known as a solid angle, is a measure of the amount of space an object occupies in three-dimensional space relative to a given point. It is similar to a 2D angle, but instead of being confined to a flat plane, it spans a three-dimensional volume.

In two dimensions, an angle is formed by two lines that intersect at a point, and the angle is the amount of rotation between the two lines. In three dimensions, a 3D angle involves three or more planes, or surfaces, intersecting at a point, and the angle measures how much of the 3D space is "enclosed" by those surfaces.

The unit of a 3D angle is called the steradian (sr). A full sphere around a point is considered to subtend 4π steradians, and any smaller portion of that sphere will subtend a fraction of 4π steradians. The steradian is the solid angle subtended at the center of a sphere by a portion of the sphere's surface area.

Mathematically, a solid angle Ω is defined as:

Ω = A / r²

Where:

A is the area of the surface portion of the sphere that is being measured.
r is the radius of the sphere (the distance from the point to the surface).
Thus, a 3D angle quantifies the extent of the region that an object covers as seen from a particular point in space.