Finding the dihedral angle of a octahedron I am trying to find the dihedral angle of a regular octahedron. All the triangles on a octahedron are equilateral triangles. So, when I drop the perpendicular from the top of the octahedron to the square base, the point it hits should be the midpoint of any of the triangles. Not sure if I am going the right direction here.
 A: Choose the vertices for the octahedron to be the six points
$$ \begin{array}{c}
(\pm 1, 0, 0) \\
(0, \pm 1, 0) \\
(0, 0, \pm 1)
\end{array} $$.
Write down the vectors for a pair of edges of a given face.  For example, for the face in the first octant, you could calculate
$$ \begin{align}
\vec{v}_1 &= (0, 1, 0) - (1, 0, 0) = (-1, 1, 0) \\
\vec{w}_1 &= (0, 0, 1) - (1, 0, 0) = (-1, 0, 1)
\end{align} $$
Now, find a vector normal to the face by calculating the cross product $\vec{n}_1 =  \vec{v}_1 \times \vec{w}_1 = (1, 1, 1) $.
Do the analogous calculation for an adjacent face to find its normal $\vec{n}_2$.  Now, calculate the angle between these normal vectors, using
$$ \cos \theta = \frac{\vec{n}_1 \cdot \vec{n}_2}{||\,\vec{n}_1\,||\;||\,\vec{n}_2\,||} $$
Convince yourself (really do it!) that the dihedral angle is the supplement to this angle.  Namely, $\pi - \theta$.
A: Yes, your direction is correct but you left it off too soon.
Dihedral angle can be next found directly by drawing a 3D diagram of the top half of the Octahedron. Take   equilateral triangle $ABV$ to find altitude $h$ by the Pythagoras thm:

From center point $M$ of $AB$ the vertex $ V$ and center of base $O$ draw a right triangle (yellow).
$$ h^2 = \big( \sqrt3\big)^2 -1^2 \rightarrow h=\sqrt2 $$
Semi-dihedral angle is
$$ \angle OMV  = \tan^{-1}\sqrt2= \cos^{-1}\dfrac{1}{\sqrt3} =\sin^{-1}\sqrt{\dfrac{2}{3}} $$
and the dihedral is double this.
A: Fix a vertex $v$ in octahedron $O$ of edge length $l$ where $l$ is
suitably large. If we place $v$ at origin, then consider
intersection of unit sphere $S$ and $O$ That is we have spherical
four polygon $P=ABCD$ on $S$ : Since face in $O$ is regular triangle
so edge length of $P$ is $\frac{\pi}{3}$ And note that internal
angle of $P$ is dihedral angle $\theta$.
If two diagonals $AC,\ BD$ intersect at $E$ then consider a
spherical triangle $EAB$. $\angle AEB=\frac{\pi}{2}$ and
$L:=|AE|=|BE|$ Hence cosine law implies that $$ \cos\
\frac{\pi}{3}=\cos^2L +\sin^2L\cos\ \frac{\pi}{2} $$ $$ \cos\
L=\cos\ L\cos\ \frac{\pi}{3} +\sin\ L\sin\ \frac{\pi}{3}\cos\ \angle
BAE $$
$$ L=\frac{\pi}{4},\ \theta=2\angle BAE=2\cos^{-1}\frac{1}{\sqrt{3}}
$$
