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How many interior diagonals does an icosahedron have? (An $\emph{icosahedron}$ is a 3-dimensional figure with 20 triangular faces and 12 vertices, with 5 faces meeting at each vertex. An $\emph{interior}$ diagonal is a segment connecting two vertices which do not lie on a common face.)

 Mar 13, 2020
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How many interior diagonals does an icosahedron have?

(An icosahedron is a 3-dimensional figure with 20 triangular faces and 12 vertices, with 5 faces meeting at each vertex.

An interior diagonal is a segment connecting two vertices which do not lie on a common face.)

 

Icosahedron :

\(\begin{array}{lcll} F=faces = 20 \\ V=vertices=12 \\ E=edges=\ ? \end{array} \)

 

For convex 3-dimensional polyhedra Euler's formula: \(\mathbf{V + F - E = 2}\) or 

\(\begin{array}{|rcll|} \hline E &=& V+F-2 \\ E &=& 12+20-2 \\ \mathbf{E} &=& \mathbf{30} \\ \hline \end{array}\)

 

Interior diagonals :

\(\dbinom{\text{vertices}}{2} - \text{edges} \quad | \quad \text{vertices}=12,\ \text{edges} = 30\)

 

\(\begin{array}{|rcll|} \hline && \mathbf{\dbinom{\text{vertices}}{2} - \text{edges}} \quad | \quad \text{vertices}=12,\ \text{edges} = 30 \\\\ &=& \dbinom{\text{12}}{2} - 30 \\\\ &=& \dfrac{12}{2}*\dfrac{11}{1} - 30 \\\\ &=& 66 - 30 \\ &=& \mathbf{36} \\ \hline \end{array}\)

 

An icosahedron has 36 interior diagonals

 

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 Mar 13, 2020

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