# Angles in 3 Dimensions

Consider the following points;

A(0,-4,4)

B(0,4,4)

C(0,-4,0)

D(0,4,0)

E(x,y,0)

If all of these are connected, it leaves a rectangular-based pyramid with a variable vertex (E). Is there any way to compare the angle at this vertex as x and y or are angles limited to a single plane involved in the shape? TO put this into a bit more context, the closer E is to the origin, logically, this 'angle' or measurement would be larger than if E was farther away.

Best Regards,

Yazan

• Excuse me but what angle of vertex (E) do you want to compare? We are in a 3D, do you want yo compare a 2D angle? – Ching-Ting Wu Nov 26 '16 at 15:43
• There's a formula here: en.m.wikipedia.org/wiki/Solid_angle. Look for Pyramid and browse till the end of the paragraph. – N74 Nov 26 '16 at 17:22

In the point $E$ are concurrent four sides of the pyramid $\overline{EA}$, $\overline{EB}$, $\overline{EC}$, $\overline{ED}$.
If you want find the angles between any two of them you have to find the vectors parallel to the sides, e.g. $$\vec {EA}=(-x,-4-y,4)^T \qquad \vec {EB}=(-x,4-y,4)^T$$ than the angle $\theta$ between them is given can be found by means of the dot product:
$$\theta=\arccos\left(\frac{\vec{EA}\cdot \vec{EB}}{|\vec {EA}||\vec{EB}|} \right)$$
• You can add the four angles and note that the sum has its maximum value $2\pi$ when $E$ is the plane of the basis. – Emilio Novati Nov 26 '16 at 17:16
• I've added a figure to my answer. The plane of the basis is the plane that contain $A,B,C,D$ , that is the plane $x=0$. You can see that if $E$ is a point on this plane (i.e. the height of the pyramid is null) than it is a poin on the side $CD$ and the sum of the angle with the common vertex at $E$ is $360°$. – Emilio Novati Nov 27 '16 at 16:33