This can be computed directly from cross-sections of the polyhedra, as was done in the homework.
But it also can be seen from the fact that tetrahedra and octahedra nest together . Specifically, a tetrahedron with side 2 can be decomposed into 4 tetrahedra with side 1 in the corners and an octahedron with side 1 in hole in the center.
The two dihedral angles can be seen in the plane as the angles of intersection of the diagonals of a special rectangle. This is the rectangle whose vertices are the endpoints of two diagonals of a cube. So the dimensions of the rectangle are 1, sqrt 2, with the length of the diagonal = sqrt 3.