Twisting a Prism

Geometry Level 5

A certain kind of prism has one square face (shown on the bottom) and one equilateral triangle face (shown on top), both with an area of 1, parallel to each other and sharing a common axis (red) perpendicular to both and passing through the centers of both. The faces are separated by a distance of 1. The edges connect the vertices of both the square and the equilateral triangle, as shown in the graphic. One of the altitudes of the equilateral triangle is parallel with one of the diagonals of the square. The volume of this prism is greater than 1.

A typical cross section of this prism parallel to the faces is an irregular heptagon, as shown in dark blue, this one being exactly halfway between the ends.

If the equilateral triangle face is rotated about the red axis by a certain angle (either way) while the square face remains stationary (and both faces remain parallel), the volume of the prism is reduced. When this volume becomes exactly 1, the same as a unit cube, this angle of twist can be exactly expressed as follows:

\[\theta =\arcsin\left( \sqrt { a } -b \right) ,\] where \(a\) and \(b\) are positive integers with \(\sqrt a> b\). Find \(a+b\).

Note: Assume that the blue edges stretch or shrink as the prism is twisted, so that all the faces of the prism stay flat.


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