Quandaries
and Queries 

I have included a drawing to help explain the problem. Once the triangle is cut out and looked down upon side A is 13cm B&C 14cm. For all 6 triangles to join flush together there needs to be an angle cut on D&E, this is the angle i can't figure out properly. Fig.1 is the finished product. Fig.2 single triangle. Fig.3 two triangles joined together not cut. Fig.4 two triangles joined with angle cut on either side. 

Hi Steve, This is a nice project. We would like to see it when it is finished. We are going to use vectors to find the angle you need. The line diagram below shows two of the triangles, PQR and PRS, and the circle with center at the center of the hexagon and radius 13. This circle passes through the six vertices of the hexagon. The height of the pyramid has been extended to reduce the clutter on the diagram.
We have overlaid a coordinate system with the origin at the center C of the hexagon, the X and Y as shown and the Z axis vetrical through P. The cordorinates of the labeled points are then
Let u be the vector from CQ, v be the vector from CR, w be the vector CS and t be the vector CP. With this notation
Vectors n_{1} and n_{2}, perpendicular to the triangles PQR and PRS respectively can be found using the cross product.
The dot product of the vectors n_{1} and n_{2} is given by
where theta is the angle between n_{1} and n_{2} and hence
Thus
The angle between the planes containing the triangles PQR and PRS is then
Thus, using your Fig. 4
Thus the bevel angle is ^{155.81}/_{2} = 77.9 degrees.
Chris and Harley 

