



 
Soren, I drew a (very crude and not to scale) diagram of what I think is the design of your birdhouse.
then drew a cross section showing a side and a piece of the roof. Is the angle A the angle you want?
Harley Soren wrote back.
Hi Soren, I think I understand now. I used vectors to approach your problem and Mathematica to perform the calculations. The cross product of the vectors v_{1} and v_{2} gives a vector n_{1} which is perpendicular (normal) to the roof section containing the vectors v_{1} and v_{2}. Likewise the cross product of the vectors v_{3} and v_{4} gives a vector n_{2} which is perpendicular (normal) to the roof section containing the vectors v_{3} and v_{4}. The angle between the adjacent roof sections is then the angle between the vectors n_{1} and n_{2} and thus the angle of the cut will be half this angle. To find the vectors I am going to impose a coordinate system with the origin at the initial point of v_{1} and v_{2} , with v_{1} along the xaxis, the yaxis pointing into the page and the zaxis vertical. Here is a cross section of the birdhouse at the base of the roof. v_{1} = OP and v_{3} = PQ. I let s be the length of a side of the octagon (the distance from O to P) and h be the height of the roof section of the birdhouse. The coordinates of O are (0,0,0) and the coordinates of P are (s,0,0). The angle PCO is 360/8 = 45 degrees or π/4 radians. (I am going to use radians for the angle measurements sinceMathematica uses radians.) The sum of the angles in a triangle is 180 degrees (π radians) and the triangle COP is isosceles, hence the angle COP is 3/8 π radians. D is the midpoint of OP and the angle ODC is a right angle and hence tan(COD) = DC/OD. Thus the ycoordinate of C is
Thus the coordinates of the point at the peak of the roof are (s/2, s/2 tan( 3/8 π), h). Triangle QPT is a right triangle with angles QPT and TQP each measuring π/4 radians (45 degrees) and PQ = s and thus PT = TQ = s/√2 and hence Q has coordinates (s + s/√2, s/√2, 0). From this information I can find the vectors I need
This will allow me to find n_{1} and n_{2} . The dot product of n_{1} and n_{2} is given by
where n_{1} is the length of n_{1} , n_{2} is the length of n_{2} and theta is the angle between them. Thus
Now I am ready to use Mathematica. Below is the output from my Mathematica session.
Cross is the cross product function, nor is a function to calculate the length of a vector in 3space, and c2t is the cos(theta) function defined above. I then calculated this function for the values you gave, s = 7 inches and h = 4 inches. The arccos function gave an angle of 0.176516 radians which I then converted to 10.1136 degrees. Thus you need to set your saw blade to 10.1136/2 = 5 degrees. Harley  


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