https://artofproblemsolving.com/wiki/api.php?action=feedcontributions&user=Pranavaddanki&feedformat=atom AoPS Wiki - User contributions [en] 2022-06-29T19:24:37Z User contributions MediaWiki 1.31.1 https://artofproblemsolving.com/wiki/index.php?title=Euler%27s_Polyhedral_Formula&diff=138950 Euler's Polyhedral Formula 2020-12-03T03:31:49Z <p>Pranavaddanki: /* Problem */</p> <hr /> <div>Let &lt;math&gt;P&lt;/math&gt; be any [[convex]] [[polyhedron]], and let &lt;math&gt;V&lt;/math&gt;, &lt;math&gt;E&lt;/math&gt; and &lt;math&gt;F&lt;/math&gt; denote the number of [[vertex|vertices]], [[edge]]s, and [[face]]s, respectively. Then &lt;math&gt;V-E+F=2&lt;/math&gt;.<br /> <br /> ==Observe!==<br /> Apply Euler's Polyhedral Formula on the following polyhedra:<br /> <br /> &lt;math&gt; \begin{tabular}{|c|c|c|c|}\hline Shape &amp; Vertices &amp; Edges &amp; Faces\\ \hline Tetrahedron &amp;4 &amp;6 &amp; 4 \\ \hline Cube/Hexahedron &amp; 8 &amp; 12 &amp; 6\\ \hline Octahedron &amp; 6 &amp; 12 &amp; 8\\ \hline Dodecahedron &amp; 20 &amp; 30 &amp; 12\\ \hline \end{tabular} &lt;/math&gt;<br /> <br /> ==Problem==<br /> 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 oe 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)<br /> <br /> == See Also ==<br /> <br /> * [[Euler characteristic]]<br /> <br /> {{stub}}<br /> <br /> [[Category:Theorems]]<br /> [[Category:Geometry]]</div> Pranavaddanki