Roof Valleys

Trigonometry, once you understand the basics, is fairly easy to use.  The sine curve/cosine curve model works great for electrical engineering, but isn’t very useful for building math.  Really, just the ++ quadrant of a unit circle is sufficient for every trigonometry problem you’re likely to encounter as a carpenter.

This is the intersection of an 8/12 roof and a 5/12 roof.  The cricket has an 8/12 slope on one side, and a 5/12 on the other.  Two of the valleys are regular, they are at an angle of 45º to plan.  The other two valleys are irregular, one side is 5/12, the other 8/12, and the valleys lie at an angle other than 45º to plan.  This requires a slightly more sophisticated approach than the conventional solutions.

Here’s a simple isometric of the basic idea, with the individual triangles lined out in colors.  The trick is to determine a common…