It will be observed that in one case the square is laid across the edge with 12" on the tongue and 12" on the blade, [Fig. 74-a]. This, as might be supposed, is for finding the cheek or side cut for jack, hip, or valley where the junction angle is 45. In the case of the octagonal hip, valley, or jack 5" must be taken upon the blade, since that is the tangent value of 22½°, 12" as base being taken on the tongue, [Fig. 74-b]. This tangent value will vary, then, according to the change in the junction angle.
Fig. 75-a.
Fig. 75-b.
Securing Value of A-B of [Fig. 74], Various Angles]
The reason for using the tangent and run for this work is indicated by the position of the square on the plan of the roof, [Fig. 75.] These figures are for use only when the timbers lie in the plane of the plate, or any parallel plane. When rafters take on pitch or rise, however, the upward projection of the plan of the miter cut, [Fig. 74], will determine the side cut as just described.
36. Rafter Lengths of Octagonal and other Polygonal Hips and Valleys.—First Method: Knowing the run of a hip or valley for the polygon under consideration (17" for the square, 13" for the octagon, etc.), by assuming the respective rises for the various pitches and solving c′² = a′² + b²', [Fig. 76], data pertaining to hip or valley unit rafter lengths, such as that for the octagon in [Fig. 73], is obtained.
To determine a rafter length, having available such a table, multiply the hip or valley length per foot of run of common rafter as given in the table by the run of the common rafter of that roof. Reduce to feet. Such lengths will be laid off by measurement from the side or cheek cut, which will have been laid off, down the top edge of the rafter.
