Problem:
A right prism with height has bases that are regular hexagons with sides of length . A vertex of the prism and its three adjacent vertices are the vertices of a triangular pyramid. The dihedral angle (the angle between the two planes) formed by the face of the pyramid that lies in a base of the prism and the face of the pyramid that does not contain measures . Find .
Solution:
Label the other vertices of the pyramid , and , where and are sides of a base of the prism. The dihedral angle of is formed by faces and . Let be the midpoint of . Then . Because , it follows that and . Therefore in right . Hence .
The problems on this page are the property of the MAA's American Mathematics Competitions