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In Geometry, What Is an Octahedron?

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  • Written By: Mary McMahon
  • Edited By: Nancy Fann-Im
  • Last Modified Date: 26 November 2016
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    Conjecture Corporation
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An octahedron is a three-dimensional solid with eight faces, each comprised of a polygon. There are 257 known configurations for convex polyhedra, with a variety of faces including triangles and hexagons. These shapes are a topic of interest in geometry and some other branches of mathematics, and they can also be important for activities like developing new packaging designs.

Polygons all contain straight lines joined together in a closed shape. The lines do not intersect with each other at any point in the shape. Some examples of well-known polygons include triangles, squares, and octagons. These shapes are named by the number of sides they have, just as three dimensional polyhedra are known by the number of faces they contain. Thus, the name “octahedron” implies that the shape has eight faces, just like a nonahedron has nine faces.

In a regular octahedron, the shape has eight equilateral triangles as the eight faces. The shape looks like two pyramids stacked base to base. One use for regular octahedrons is in the creation of eight sided dice. These dice are used in some specialty games where players want more than six options to come up when they roll dice. It is also possible to find dice with an even greater number of faces, all of which are regular polyhedra to ensure that they roll evenly and reliably.

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A two-dimensional diagram of a polyhedron that shows all the faces and how they connect is known as a net. Nets for octahedra can demonstrate the myriad of ways that eight polygons can be arranged to form a solid shape. These can include symmetrical constructions like regular octahedra and hexagonal octahedra, as well as more irregular shapes where the faces are of different sizes and shapes.

Finding the volume of a convex octahedron is a relatively simple task with many shapes. It may be necessary to break the shape down into simpler structures like pyramids to calculate their volume and add them together. Concave octadehra are more difficult to work with, as the faces that cut in can complicate volume measurements. Formulas are available to help people solve questions of volume quickly, especially for standardized shapes like the regular octahedron.

The octahedron is sometimes used in product packaging. While it is not always the most efficient shape, it can be visually interesting and for some applications it can help package oddly shaped items in the most effective and safe way. These shapes are also used in the construction of toys, some of which may break apart to allow children to explore different configurations of their faces.

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