With these, we have to find the number of faces of the octahedron. Paul Kunkel email to reach me, the word geometry must appear in the body of the message.From the question, we were given the number of vertices on the octahedron and the number of edges of the octahedron. Together, the octahedron and tetrahedron pack space, but neither of them can do so alone. It is ready for another layer of tetrahedra. Now fit twelve octahedra around this figure. It might also be visualized as two intersecting tetrahedra. Attach a tetrahedron to each of its eight faces. Start over again with a single octahedron. Edges vanish, and it becomes a heptahedron, having three rhombic faces and four triangular faces. Less obvious is the fact that three other pairs of faces are coplanar. This follows from the fact that their dihedral angles are supplementary. It is possible to lay a tetrahedron on the same surface and nestle it right underneath the octahedron. Lay an octahedron on a horizontal surface. This fact follows naturally from its dual relation with the cube.Ī cross-section of the octahedron can be a square or a regular hexagon.Ī planar projection also may be a square or a regular hexagon. Do the same to a cube, and the result is an octahedron.Ī tetrahedron can be formed by connecting centers of certain faces of the octahedron. Connect the centers of adjacent faces, and the result is a cube. Each has a unit square for a base and a height equal to the circumradius. Next, find the apothem of an equilateral triangle, and use it to find the inradius and circumradius of the octahedron.Īnother way with the volume is to dissect the octahedron into two pyramids. The dihedral angle between the two triangles is the dihedral angle of the octahedron. At each of the base vertices there are three faces, including two triangles and a square. The result is a pyramid with a square base. Four faces that meet at a common vertex are cut away from the rest of the solid.

It applies only to cases in which exactly three faces meet at a vertex. There is a problem with the dihedral angle formula being use in these pages. Multiply that by the number of faces for the total surface area. The octahedron might also be classified as a square dipyramid or a triangular antiprism.Ĭonsider an octahedron with side length s. The octahedron has 8 equilateral triangular faces, 6 vertices, and 12 edges.

Octahedron faces edges vertices