Difference between revisions of "Euler's Polyhedral Formula"
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Apply Euler's Polyhedral Formula on the following polyhedra: | Apply Euler's Polyhedral Formula on the following polyhedra: | ||
| − | <math> \begin{tabular}{|c|c|c|c|}\hline Shape & Vertices & Edges & Faces\\ \hline Tetrahedron &4 &6 & 4 \\ \hline Cube/Hexahedron & 8 & 12 & 6\\ \hline Octahedron & 6 & 12 & 8\\ \hline Dodecahedron & 20 & 30 & 12\\ \hline \end{tabular} </math> | + | <math> \begin{tabular}{|c|c|c|c|}\hline Shape & Vertices & Edges & Faces\\ \hline Tetrahedron &4 &6 & 4 \\ \hline Cube/Hexahedron & 8 & 12 & 6\\ \hline Octahedron & 6 & 12 & 8\\ \hline Dodecahedron & 20 & 30 & 12\\ \hline Icosahedron & 12 & 30 & 20\\ \hline \end{tabular} </math> |
==Problem== | ==Problem== | ||
| − | A convex polyhedron has for its faces 12 squares, 8 regular hexagons, and 6 regular octagons. At each vertex of the polyhedron one square, one hexagon, and | + | A convex polyhedron has for its faces 12 squares, 8 regular hexagons, and 6 regular octagons. At each vertex of the polyhedron one square, one hexagon, and one octagon meet. How many segments joining vertices of the polyhedron lie in the interior of the polyhedron rather than along an edge or a face? (1988 AIME #10) |
== See Also == | == See Also == | ||
Latest revision as of 13:15, 6 March 2022
Let 
be any convex polyhedron, and let 
, 
and 
denote the number of vertices, edges, and faces, respectively. Then 
.
Observe!
Apply Euler's Polyhedral Formula on the following polyhedra:
Problem
A convex polyhedron has for its faces 12 squares, 8 regular hexagons, and 6 regular octagons. At each vertex of the polyhedron one square, one hexagon, and one octagon meet. How many segments joining vertices of the polyhedron lie in the interior of the polyhedron rather than along an edge or a face? (1988 AIME #10)
See Also
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