# Dihedral angles for expanded icosidodecahedron.

• StevenM
In summary, the polyhedron in question is formed by adding squares and golden rhombi to an icosidodecahedron. The dihedral angle between a pentagon and a square is 162°, and the angle between a square and a triangle is 142.62° before the squares are added. The polyhedron can be viewed on the website provided if the 3D model cannot be accessed.
StevenM
I am trying to determine two dihedral angles for this polyhedron:

http://georgehart.com/virtual-polyhedra/vrml/zonish-6-icosidodecahedron.wrl

1. The angle between a pentagon and a square?
2. The angle between a square and a triangle?

This polyhedron is formed by adding squares between the pentagons and triangles of an icosidodecahedron, and adding golden rhombi where each of the vertices are.

This polyhedron can also be viewed at the top of this page if you do not have the plugin to view the 3D model:

http://www.georgehart.com/virtual-polyhedra/zonish_polyhedra.html

I know that the dihedral angle between a rhombus and a square is 162°, and that in an icosidodecahedron the dihedral angle is 142.62° between the pentagons and triangles before the squares are added. Any help would be appreciated.

Nevermind. I figured it out. :)

## What is a dihedral angle?

A dihedral angle is the angle between two intersecting planes. In the context of geometry, it refers to the angle formed between two adjacent faces of a polyhedron.

## How do you calculate dihedral angles for an expanded icosidodecahedron?

To calculate the dihedral angles for an expanded icosidodecahedron, you can use the formula: θ = arccos(-1/3), where θ is the dihedral angle. This formula is derived from the fact that an expanded icosidodecahedron has 12 pentagonal faces and 20 hexagonal faces.

## What is the significance of dihedral angles in geometry?

Dihedral angles play an important role in geometry as they help determine the shape and symmetry of a polyhedron. They are also used in various applications such as crystallography and structural engineering.

## How do dihedral angles affect the stability of a polyhedron?

In general, larger dihedral angles indicate a more stable polyhedron as they distribute the forces applied to the faces more evenly. However, this can vary depending on the specific shape and structure of the polyhedron.

## Are there any real-life examples of objects with dihedral angles for an expanded icosidodecahedron?

An expanded icosidodecahedron is a theoretical shape and is not commonly found in everyday objects. However, it is a common shape used in puzzles and games, such as the Rubik's Cube and the Magic 120-cell puzzle.

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