Say I have a cone where I have 3D slice of it running from the apex to the base. The edges of the slice meet at the apex at a $150°$ angle. We are assuming that we know the lengths of the slice's edges and that lengths of both of these edges are equal. Given these details, how would I be able to find the length of the section of the cone's base that is part of the slice?
Here is an image to sorta show what I am referring to. The edges of the slice are shown by the two dark gray lines. The segment of the edge of the cone's base that is in between the dark gray lines is what I am trying to find. The $150°$ angle is located at the apex of the cone, in between the lines. NOTE: image is not to scale.
Let us assume that the base of the cone is a unit circle and the height is $h$. The requested angle (equal to the arc length) is denoted $\theta$, while the given angle between the apothems is $\phi$.
Let the apex of the cone be at $(0,0,h)$, and the feet of the apothems $(1,0,0)$ and $(\cos\theta,\sin\theta,0)$. We express the angle $\phi$ by the dot product of two unit vectord
$$\cos\phi=\frac{(-1,0,h)}{\sqrt{1+h^2}}\cdot\frac{(-\cos\theta,-\sin\theta,h)}{\sqrt{1+h^2}}.$$
Hence
$$\cos\theta=(1+h^2)\cos\phi-h^2.$$