As we have found, the run of a hip rafter equals the diagonal of a square formed by the run of the common rafters. Therefore, returning to the 24' house mentioned in B, Topic 50, we find that the run of the common rafter is 12' 0", the diagonal of which will be a constant applicable to the runs of all hip rafters, as 12 is the constant of the run of all common rafters.

With the usual mathematical formula for finding the length of the hypotenuse of a right-angled triangle, we will find the desired constant for the run of the hip.

B = base. A = altitude. H = hypotenuse.

Formula 21. H =

= √288 = 16.971 feet.

This means that the run of the hip rafter of a 24/ house is 16.971', but in practice it is customary to use 17 for the constant, as it is near enough for all practical purposes.

As the rise of the roof is the same for a hip rafter as for a common rafter, it is obvious that the constant used for the rise of the common rafters will be used for the rise of the hip.

Thus the constant for the run of the hip and valley rafters of any rectangular house of even pitches will be 17, and the constant for the rise of the hip and valley rafters will be 12, 8, or 6, as the roof is half, third, or quarter pitch.

A = 12, 8, or 0, the constant of the rise of the hip rafter.

P = plumb cut.

S = seat cut.

R1= 17, or the run of the hip rafter.

Formula 22. 17 on Bl., A on To., Bl. = S, To. = P.

(D.) Having found the plumb and seat cuts, we must next find the side or cheek cuts, which should be made at the same time that the ridge or plumb cut is made, as the rafter fits against the ridge, which it intersects in the plan at an angle of 45°, upon an ordinary roof, as shown at b, Fig. 77. The simplest method of obtaining this cut, for a square house, is to lay off the plumb cut, ab, of a, Fig. 79, upon the side of the hip rafter; measure the thickness of the rafter, c, or 2" parallel with the plumb or ridge cut, and draw the line de. Square across the top edge of the rafter to the other side, locating the point f; draw a line from f to a diagonally across the top edge of the hip, obtaining thus the desired cheek or side cut.

The student should study this problem carefully, as it involves the principle by which all side or cheek cuts are made, according to the method taught in this book. It will be seen by the plan of the roof that the horizontal angle of the intersection of the hip rafter and the ridge is an angle of 45°; the fact that the center of the roof is higher than at the plates does not alter in the slightest degree that angle. Therefore in taking off the cheek cut equal to the thickness of the rafter we are making what is simply a square miter joint, when the horizontal section is considered.

Fig. 79. - Method of finding The Cheek Cuts.

In order to assist the student to a better understanding of this problem, let him draw a diagram similar to b, Fig. 79. Square across the hip rafter from the short corner a to the long side, at b, and measure from b to c. This will give the distance which should be measured back parallel to the ridge or plumb cut of the rafter, as at c.

In this case it will be the thickness of the rafter, but if the angle of contact between the hip and the ridge was any other angle, the distance bc would vary accordingly, as indicated by the dotted lines. This method may be applied to any angle of intersection where a cheek cut is necessary.

In cutting hip rafters the top end usually is cut first, and the length taken from the center of the side or cheek cut, upon the top edge, as at b, Fig. 77. Point g indicates the exact length, and the distance f shows the horizontal allowance which must be made for one half of the thickness of the ridge.

(E.) The graphic method of laying out the cuts of hip rafters is described in Fig. 80, and is sometimes used upon intricate roofs, to prove the angles found by the steel square, though its principal use is in solving problems in roof construction which are published in the periodicals that circulate among carpenters.

In Fig. 80, the angle a is the plumb or ridge cut, and b the seat cut of a common rafter of a third pitch roof.

Those may be found by drawing the run, cb, and the rise, ca, as shown, and connecting them by the line ab, which indicates the top. or pitch of the rafter.

The roof plan of a house 24' wide of any pitch may be drawn to a scale and the bevels found; these will be the same in all rectangular houses of the same pitch.

In ascertaining the length, de, the plumb cut, e, and the seat cut, d of a hip or valley rafter, the same graphic method may be followed, the results being shown at e and d. In order to obtain the cheek, or side cuts of the hip, valley, jack, and cripple rafters, set compasses at the radius de, and, with d as center, draw an arc cutting the ridge at f1; connect d and f1. The angle f1 is the cheek cut of all of the rafter ends marked f. The top end of the valley rafter (gh) is cut the same as the plumb cut of a common rafter of the same rise and run. The plumb cut at the plate or at the end, h, of the valley rafters should be cut as shown at a, in Fig. 81, to allow them to fit into the inside angle of the plate.

Fig.80. -The Graphic Method of finding the Lengths and Angles of Rafters.

Fig. 81. - Plate Cut Of Valleyrafters.

(F.) Backing a hip rafter is illustrated by a, Fig. 82.

The student should understand that as the rafter is measured in the center of the top edge, which is the line of the hip, the corners of the upper edge of the hip will project above the line of the common rafters, leaving a triangular space, as at b, if the plumb height at the plate is the same as that of the common rafters. The process of laying out and beveling the top of a hip rafter, as at a, so that the roof sheeting, or sheathing, will lie perfectly flat, and meet that of the other side of the hip, directly over the line of the hip, as at a, is called backing. The following formula will give the necessary bevel for the backing, or the amount which is to be cut off, and which should be laid off on the top of the rafter; in this case, a half pitch rafter is illustrated.

Transfer the distance, a, Fig. 83, to the side of the rafter, as shown at b. The wood between the center of the rafter on the top edge and the distance a on the side should be cut away.

Fig. 82. a, Backed Hip Rafter; b, Square Hip Rafter.

Fig. 83. - Method of laying out the Backing.

B = backing.

R3= run of common rafter.

A = rise of the roof.

Formula 23. B = R3 on Bl., A/2 on To. To. = B.

Hip rafters are rarely hacked upon common work, as there is not enough gained to make it advisable. As a substitute, the plumb height of the rafter at the plate may be shortened a distance equal to the height of the backing (a, Fig. 83), which should be found by the above method. After the rafter is in place, the sheeting may be nailed across to meet that of the other side of the roof. This will leave a triangular hole between the sheeting and the top of the hip rafter, as shown at b of Fig. 82.

This method is perfectly satisfactory for common work, as the nails are driven into the corners of the hip which are flush with the common rafters.