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?
Soren wrote back.
I think I understand now.
I used vectors to approach your problem and Mathematica to perform the calculations.
The cross product of the vectors v1 and v2 gives a vector n1 which is perpendicular (normal) to the roof section containing the vectors v1 and v2. Likewise the cross product of the vectors v3 and v4 gives a vector n2 which is perpendicular (normal) to the roof section containing the vectors v3 and v4. The angle between the adjacent roof sections is then the angle between the vectors n1 and n2 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 v1 and v2 , with v1 along the x-axis, the y-axis pointing into the page and the z-axis vertical.
Here is a cross section of the birdhouse at the base of the roof. v1 = OP and v3 = 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 y-coordinate 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 n1 and n2 .
The dot product of n1 and n2 is given by
where |n1| is the length of n1 , |n2| is the length of n2 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 3-space, 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.
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