Octahedrons are three-dimensional shapes made up of eight triangular faces. These geometrical shapes are one of the five Platonic solids. Octahedra have 8 faces, 12 edges, and 6 vertices. Four faces of the octahedron meet at each vertex.
Here, we will learn more about the faces, vertices, and edges of octahedrons. We will use diagrams to visualize these parts.
Faces of an octahedron
The faces of octahedra can be considered as the flat surfaces that form the three-dimensional shape of the octahedron. Therefore, we can determine that an octahedron is made up of eight triangular faces.
Alternatively, we can also consider the faces of any 3D shape as the two-dimensional figures formed by the vertices and the edges.
Octahedra can be formed by joining two square pyramids at their bases. Each pyramid contributes four side faces for a total of eight. The faces of each pyramid meet at the top vertex, which can be considered the main vertex.
When we talk about octahedrons, we usually mean a regular octahedron. The main characteristic of the faces of these octahedrons is that they are equilateral triangles. That is, their three sides have the same length.
If we want to calculate the surface area of an octahedron, we simply have to find the area of one of the faces and multiply by 8. Therefore, we have:
$latex A_{s}=8A_{t}$
where, $latex A_{t}$ is the area of one of the triangular faces.
Alternatively, we can calculate the surface area of a regular octahedron, using the standard formula:
$latex A_{s}=2\sqrt{3}~{{a}^2}$
Vertices of an octahedron
The vertices of any three-dimensional geometric shape are the points where two or more edges meet. In the case of octahedrons, the vertices are the points where four edges meet.
Alternatively, we can also consider the vertices of an octahedron to be the points where four faces of the octahedron meet.
In total, we have 6 vertices in every octahedron. 1 vertex is located at the top, 1 at the bottom, and 4 exactly in the middle of the octahedron. The vertices in the middle form a square.
The upper and lower vertices, where four lateral faces meet, can be considered as the main vertices of the octahedron, since they contain the axis of symmetry.
Edges of an octahedron
The edges of any three-dimensional figure can be considered as the line segments joining two vertices. The edges form the outline of each face.
Another way to define edges is as the line segments where two triangular faces of the octahedron meet. This means that the edges are located on the limits of the octahedron.
In total, we have 12 edges in every octahedron. 4 edges meet in the upper pyramid, 4 in the lower pyramid, and 4 exactly in the middle of the octahedron. The edges in the middle form a square.
In the diagram, we see that each face of the octahedron has three edges, and each edge is shared by another face.
See also
Interested in learning more about octahedra? Take a look at theses pages:
Jefferson Huera Guzman
Jefferson is the lead author and administrator of Neurochispas.com. The interactive Mathematics and Physics content that I have created has helped many students.
