THEORY: Hip and Valley Roof Trigonometry

    The focus is on concise, generic trigonometric methods of resolving the angles required for accurate joinery in a Hip / Valley roof. Diagrams are used extensively to supplement the abstract trigonometry. The following ideas are explored:

Section of hip roof, with angles in their standard locations relative to each other

All kernels are equivalent (Cognate kernels)
The sine, cosine, and tangent of any kernel angles may be resolved in terms of two other angles
Compound angle relationships on the stick
The location of right angles on the roof, on the kernel, and on the stick
Definition, measurement and relationships between Dihedral, Saw blade, Miter and Bevel angles
General solutions of Deck angles
Utilizing a unit reference length to determine trig functions of angles
Extracting kernels from the stick
Extracting kernels from Interlocking kernels
Extracting sticks from kernels
Superimposing sticks on kernels
Models other than tetrahedral kernels

    ANGLE CALCULATORS   contains links to Excel worksheets and Javascript calculators which will solve the complex roof framing and joinery angles based on the theory in this section.

Overview of Tetrahedra and Triangles defining a Hip Roof

Deck Angles (Trigonometry).doc
Deck Angles (Trig and Geometry).pdf

Deck Angles (Geometry).doc
Deck Angles (Geometry).html
Compass and Straightedge Solutions.htm

Common Rafter Angles.doc
Common Rafter Angles.pdf

Purlin Related Angles.doc
Purlin Related Angles.pdf

Valley Angles.doc
Valley Rafter Angles.pdf

General Hip/Valley Model.doc
General Hip/Valley Model.pdf

Rafter/Purlin Square Cut Angles.doc
Chamfer (Square) Cuts on Rafters and Purlins.pdf

Valley Square Cut Angles.doc
Chamfer (Square) Cuts on Hip/Valley Rafters.pdf

Alternate Square Cut Model.doc
General Square Cut Model.pdf