To frame the remaining hips of [Fig. 83-a] proceed in a manner similar to that just described. In this case the angle made by the hip when it lies in the plane of the plate and the side of the house will be 67½° degrees. Why? [Fig. 84-a]. The framing tool or protractor and T-bevel provide the simplest means of framing. If one wishes to use the framing square he will find that by using the tangent value of 67½° he gets 28.97" to take on the blade when 12" is taken on the tongue. This he cannot find, of course. He may either take other smaller numbers having the same ratio, or he might better take the cotangent value of 67½°. Cot of an angle of a right triangle is the ratio of the adjacent side to the opposite side.
Solution:
Angle A′ = 67½°, Fig. 84-a.
Cot 67½° = .414 when b′ = 1.
When b′ = 12" cot A′ = 4.97" or 5".
Fig. 84-a. Fig. 84-b.
Framing Octagon Hip Intercepted at 67½°
(1) Place the square as in [Fig. 84-b] and scribe along the blade. By this time the student should have observed that the tangent value of any angle increases as the cotangent value of the angle decreases and vice versa. Note that when a cotangent value is used, 12" is still taken on the tongue and the same number as for the tangent value of that number of degrees is taken on the blade, but that the scribing is done along the blade and not along the tongue as is the case when tangent values are used. (2) Proceed from this point as in [Sec. 35], (2) et seq.
To frame the jack intercepted by the side of the house. [Fig. 85-a]; (1) Proceed to find the miter for the intercepted jack when it lies in the plane of the plate. It is 45 degrees. Tan 45 degrees = 1 when b″ = 1. Tan 45 degrees = 12" when b″ = 12". (2) Place the square as in [Fig. 85-b] and scribe along the tongue. (3) From this on, continue as in [Sec. 35], (2) et seq.
Fig. 85-a. Fig. 85-b.
Framing an Intercepted Jack
From these solutions the student should be able to generalize sufficiently to care for rafters of any angle of intersection with the side of a building, and of any pitch. The use of the framing tool or protractor and T-square is strongly recommended upon such work as this. Nothing but tradition prevents its more general use in carpentry. Having determined the lay-out for the side cut of these rafters, the length of each must next be determined. To determine the length of hip or jack intercepted by the side of the house and the plate: (1) Determine the run of the intercepted part of the common or jack rafter, as b, [Fig. 83-a]. (2) By means of the table of rafter lengths per foot of run compute the length of rafter under consideration.
In determining the run of an intercepted part it is necessary to have data concerning the size of the octagon and the amount cut off by the building. The lines E-D and F-D, [Fig. 83-a], when dimensioned, give this data. The actual lengths of the various hips, jacks, etc., in practical carpentry, upon small bays, are usually determined by actual measurement from the plate or sill to the proper point of intersection on the building as indicated by a stick or extension rule held at the proper angle. Sometimes a large scale drawing is made and the runs taken from this. Upon large work where accuracy is necessary and measurements impossible, trigonometric solutions should be used.