CHAPTER IVRoof Frame: any Polygon

Fig. 68. Tangents

30. Tangents; Miter Cuts of the Plate.—Before the principles involved in the laying out of rafters on any type of roof can be understood, a clearer idea of the term tangent as used in roof framing must be had. A tangent of an angle of a right triangle is the ratio or fractional value obtained by dividing the value of the side opposite that angle by the value of the adjacent side. The tangent at the plate, to which reference was made is the tangent of the angle having for its adjacent sides the run of the common rafter and the run of the hip or valley. By making use of a circle with a radius of 12" we may represent the value of this tangent graphically in terms of the constant of common rafter run,Fig. 68.By constructing these figures very carefully and measuring the line marked tangent, we may obtain the value of the tangent forthe polygon measured in inches to the foot of run of the common rafter. Such measurements, if made to the 1/100 of an inch will serve all practical purposes. A safer way, however, is to make use of values secured thru the trigonometric solutions described in Appendix I, using the graphic solutions as checks. The values of tangents at intervals of one degree are given in the Table of Natural Functions, Appendix II. By interpolation, fractional degree values may be found.

Fig. 69-a. Table of Tangents

Fig. 69-b. Rafter Table.

Example:Find the value of the tangent for an octagonal plate.Solution:Angle A′ ofFig. 68= 22½°(1/16 of the sum of all the angles about a point)Tan 22½° = .4143Tables are builded with 1 as a base. In roof framing 1" or 12" istaken as the constant or base, or unit of run of common rafter..4143 may be considered as feet, which equals 4.97".In a similar manner tangents may be found for plates of buildingsof any number of sides.

InFig. 69is illustrated a handy device one side of which, by the twirling of one disk within the other, can be made to give tangent values, in terms of a 12" base, for any number of degrees. Thereverse side of this "key" gives data to be used in the framing of square cornered and octagonal roofs. Such a key will be found a convenient way in which to carry needed data and should be easily understood and intelligently used, once the principles discussed in this chapter are mastered. An explanation of the author's key,Fig. 70, will be found in Appendix IV.

Now as to some of the uses for tangent values: First, by taking 12" on the tongue and the tangent value in inches per foot of common rafter run upon the blade of the square, we are able to get the lay-out for the miter joint of the plate.

Fig. 71-billustrates the square placed for the lay-out of the octagonal plate or sill miter. Five inches is taken as tangent since the real value 4.97" is equivalent to 5" for all practical purposes.

For the square cornered building 12" and 12" would be used in making the plate miter lay-out, since the tangent of 45 is 1 according to the Table, Appendix II. Any other like numbers would give a tangent value of 1, of course, but it is best to consider 12" on the tongue, in which case 12" must be taken on the blade.

Second, this tangent value is needed in determining the cheek or side cut of hip, valley and jack rafters, as will be shown inSec. 35.

Third, this tangent value is needed in determining the amount of backing to be given hip rafters. This is discussed inSec. 39.

Not infrequently the plate miter in degrees is required. This is determined for any regular polygon by the proposition: The plate or miter angle of any regular polygon = 90 - (central angle/2)

Example:Find the value of the plate miter of the octagon.Solution:The octagon has 8 sides; therefore central angle = 45°45° ÷ 2 = 27½°90° - 22½° = 67½°

Fourth, the tangent value is needed in finding the length of a side of a polygon, the span or run of the polygon being known, and vice versa. Length of side = span x tangent of plate, using 12" as base.

Example:An octagonal silo has a span of 18'; determine the length of platefor any side.Solution:The tangent value of the octagon = 4.97" (to each 12" of run) 18 × 4.97"= 89.46" = 7' 5.46" = 7' 5½".

Example:A side of a hexagon measures 4'; determine the run of the hexagon.Solution:Transposing the rule above: Span = length of side divided by tangentof plate.Tangent of hexagon = 6.92" when base = 12".4' divided by 6.92" = 6' 6.48" = span. Run = 3' 3.24".

31. Octagonal Roofs.—While the square cornered building is the most common, the octagon is frequently used in the form of a bay attached to the side of a house. The octagon is also common upon silos and towers. The manner of finding the run, tangent, length of hip and valley rafter, miter cut of plate or sill, the mannerof determining the numbers to use on the square to lay out the plumb and seat cuts, etc., will be found developed herein for both square and octagonal roof. Having mastered the principles involved in these two forms, the student should be able to work out framing problems for roofs of any number of sides.

32. Common Rafter for Plate of any Number of Sides.—By referring to Figs.68and72it will be seen that common rafters have for their runs the apothem of the polygon made by the plate, represented inFig. 68by the linesb, b’, b″. The run of the hip is represented by the linec, c′, c″. The rise will be found the same for full length common rafters and hips. Plumb and seat cuts and lengths per foot of run of common rafters and jacks are determined for a building of any number of sides just as for the square cornered building. The degree of inclination of common and jack rafter is applicable, too,Fig. 49.

Fig. 72. Run of Common Rafter of any Roof is Apothem of Polygon

33. Hip and Valley Rafters for Octagonal and other Polygons.—Before the table of constants for hips and valleys for octagonal and other polygonal roofs can be formed, it is necessary to determine the tangent values of these polygons, as described inSec. 30.

Proceeding with the octagon, whose tangent was found to be 4.97" when the run of the common rafter was taken as 12", by the formulac²=a²+b²,Fig. 72, the run of the octagon hip or valley will be found to be 12.99" for each foot of run of the common rafter.

34. Plumb Cut of Octagonal and other Polygonal Hips and Valleys.—The run of an octagonal hip or valley is 12.99" for each foot of run of common rafter. For practical purposes this is considered as 13". To lay off a plumb cut for an octagonal hip or valley, take 13" on the tongue and the rise per foot of run of common rafter on the blade,Fig. 73; scribe on the blade.

Fig. 73. Table for Octagon Hip or Valley

In a similar manner, having determined the tangent of any polygon under consideration, the run of hip or valley per foot of run of common rafter may be figured. The result will give the number to take on the tongue when the rise per foot of run of common rafter is taken on the blade, for laying out plumb cuts.

35. Side or Cheek Cuts of Hip and Valley Rafters for Roofs of Any Number of Sides.—Fig. 74illustrates the principles involved and method used in determining side cuts whatsoever the pitch and number of sides involved. (1) Lay the square across the jack or hip or valley, whatever is to be framed, at any convenient place, using on the blade the tangent (with 12" as base) of the polygonbeing framed, and on the tongue the constant of the run of the common rafter, 12". Scribe along the tongueA-C,Fig. 74.(2) With the square, carry the lineB-Cacross the edge as indicated. (3) Lay off the plumb cut for the required pitch, taking upon the blade the rise, and upon the tongue the constant of the run of the hip or valley or jack, according to the requirements for that particular member as determined by the style of roof. (4) Measure square back from this plumb cut line the distanceA-BofFig. 74.(5) Thru point A scribe a lineD-Aparallel to that of the plumb cut line and (6) square this across the edge as atD-E. (7) Adjust the bevel to pass thruEandF,Fig. 74.(8) Scribe a plumb cut line upon the reverse surface of the stock.

Fig. 74-a.

Fig. 74-b.

Laying off Side Cut of Jack, Hip or Valley, any Polygon]

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 theblade, 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 ofFig. 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 solvingc′²=a′²+b²',Fig. 76, data pertaining to hip or valley unit rafter lengths, such as that for the octagon inFig. 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 rafteras 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.

Fig. 76. King-Post

Lengths of hip or valley tails will be determined in a similar manner from the same table.

Second Method:This consists in successive placings of the square, using the same numbers on tongue and blade as will be used in laying out the plumb cut for this particular roof. The successive advances will be determined, as in the hip or valley of a square cornered building, by the number of feet in the run of the common rafter of this roof. A fractional part of a foot in run will be treated in a manner similar to that described for the square cornered building. Suitable allowance must be made for the fact that the length of rafter is along the middle of the top edge of the rafter, when this latter method is used.

Example:An octagonal roof of ¼ pitch has a span of 25′.The run of common rafter = 12' 6".Taking 13" on the tongue and 6" on the blade lay offsuccessively 12 measurements. Take 6/12 or ½ of 13" or 6½"on the tongue and 6/12 or ½ of 6" or 3" on the blade forthe fractional footof run.

37. Reductions in Lengths for King-Post.—Suitable reduction must be made for king-post, should there be one,Fig. 76, or for rafter thicknesses should no king-post be used,Fig. 75-b. Where a king-post is used the reduction will be ½ the width of the square out of which the king-post is formed, the measurement being made square back from the line of the plumb cut, as in reducing common rafter lengths for ridge piece,Fig. 56.

Where an apex is formed as inFig. 75-b, one pair of hips is framed each with a run equal to ½ the octagon's diagonal, with cuts at the top the same as those of common rafters. The second pair will be similarly framed but the lengths will each be reduced an amount equal to ½ the thickness of the first pair, measuring straight back from the plumb cut. The third and fourth pairs will be reduced an amount equal to ½ the diagonal thickness of the rafters already framed, measured straight back from the plumb cut, Figs.74-aand75-b. These rafters will have to have double side cuts as indicated inFig. 75-b. It will be noted that these latter rafters meet the others at an angle of 45 degrees, making the framing of the side cut similar to that of a square cornered building except that these are given double cheeks.

38. Seat and End Cut of Octagonal and other Polygonal Hips and Valleys.—The method of procedure in laying out the seat and end cuts of octagonal and other polygonal hips and valleys is similar to that described for the square hip, except of course, the numbers to use on the square will differ. These numbers will be determined by the run of the hip for that particular roof, and by the rise in inches per foot of run of common rafter. In the case of the octagon the numbers will be 13" on the tongue, and the rise on the blade. The length of tail may be figured by the tables, if such are available, or the framing square may be advanced successively as many times as there are feet in the run of the tail or the lookout of the common rafter of that roof, as heretofore described.

The miter cut on the end of a hip rafter and in the crotch of a valley rafter will vary with the tangent value of the plate. For the octagon, whose tangent is 5" the measurement from the end cut and at right angles to it will be 5/12 that used on the square cornered building. This same ratio will hold for the measurement at the crotch of the valley,Fig. 65-b.

39. Backing Octagonal and other Hips.—The principle involved in determining the amount of backing on hip rafters for the octagon, as well as that of other polygons, is similar to that for backing hips for the square cornered building.

Fig. 77. Laying out Backing for Octagon Hip

(1) Place the framing square upon the hip as for making the seat cut of the octagon hip rafter,Fig. 77.

(2) Referring toFig. 78it will be seen that the more sides a roof possesses the less will be the backing required.Fig. 78represents the hips on square, octagon, and hexagon as they would appear upon the plan if made to stand straight up at the corners of the plates. The backing in each case is determined by the tangent of the angle whose adjacent side is ½ the rafter thickness, and whose angle is equivalent to ½ the central angle asA, A′, A″,Fig. 78.The fact that the hips are inclined and not vertical in their final position in a roof makes no difference in the principle of determining backing because of the fact that our measurements are made in a horizontal plane, or in the plane of the plate, Figs.64and77.

The ratio of the backing of the octagon or any other polygonal hip to that of the square of equal rise will be proportional to their tangent values. For illustration, on the square roof we measuredalong the tongue of the square, from the top edge of the hip,Fig. 64, ½ the thickness of the rafter, for reasons made plain inFig. 78.In laying off the octagon backing we should lay off along the tongue in the line of the octagon seat cut, 5/12 of ½ the thickness of the rafter. In the case of a 2" rafter, 5/12" will be set off, using that side of the square containing graduations in 12ths.

(3) Where backing is not desired the rafter will be dropped a plumb distance equal to the plumb distance of the backing of the rafter,Fig. 77.ConsultFig. 73.

Fig. 78. Backing for Hip of any Polygon

40. Framing the Octagonal and other Polygonal Jacks.—Plumb cuts of jacks are determined by the rise and run of the common rafter of that particular roof. The seat cuts will be determined by the same numbers. Side cuts, end cuts, and lengths of jacks are determined as described in Sections 41 and 42.

41. Side Cut of Octagonal and other Polygonal Jacks.—First Method:The method described in Section 35 is applicable to jacks as well.

Second Method:If a table of common rafter lengths per foot of run is available, the side cut of any polygonal jack may be laid out by placing the framing square across the top of the rafter, taking the tangent value of the polygon on the tongue, and the length of common rafter per foot of run on the blade, and scribing on the blade. The tangent of the polygon forms the opposite sideof a triangle in which the adjacent side is the length of common or jack rafter per foot of run. The angle formed being the angle of side cut of jack.

42. Lengths of Octagonal and other Polygonal Jacks.—The methods of procedure in determining the lengths of jacks for other than square cornered buildings differs from that described for the square cornered building only in the fact that the runs, hence lengths, of the jacks must be determined differently. An examination of Figs.68and72will show that the runs, hence the lengths, of jacks for the square, the octagon, etc., when the run of common rafter is the same, will vary inversely as the tangent ratio. For example, the tangent of the square is 12" when its run of common rafter is 12", while that of the octagon is 4.97" or practically 5". The runs, and, therefore, the lengths of octagon jacks will differ one with reference to another, 12/5 of those of the jacks for the square building. If an octagon jack rests 24" from the corner, its run, and therefore its length, will be 12/5 times that of a jack on a square cornered building similarly placed.

Second Method:Difference in lengths of jacks may be determined by counting the spacings along the plate and dividing the length of common rafter by that number.

43. Framing by Means of a Protractor.—By means of a protractor used in connection with the columns containing degree measurements, Figs.49,60, and73, roof framing may be greatly simplified.

To lay off a seat cut, it is merely necessary to look in the table of hip, valley, jack, or common rafter, whatever is being framed, and read the degree of inclination of rafter for the pitch required. The blade of a T-bevel is set by means of the protractor to this angle; or a combination tool may be used, and the tool applied as inFig. 79.

The plumb cut and seat cut are complementary. Since a protractoris made to read in either of two directions, the plumb cut setting may be got by adding to or subtracting from 90 degrees the angle of inclination of that rafter. With a combination tool, one setting of the tool serves to lay out both seat and plumb cut.

Fig. 79. Laying off Seat Cut withFig. 80.Laying off Plumb Cut with Framing Tool Framing Tool

The degree of setting for a side or cheek cut, the gage setting for backing, and the amount of drop where no backing is used will be found under appropriate columns in the tables referred to above. The manner of applying the combination tool for laying out plumb cut, seat cut, side cut and end cut is indicated in Figs.79,80,81and82. The manner of determining the data found in these tables is a matter of trigonometric solution, no more difficult than that already given, and omitted for lack of space. Such problems may well form a part of the pupil's work in mathematics.

For Example: Data for side cuts of jack, hip and valley for square and octagonal roofs will be found in Figs.49,60, and73. In these tables will be found a column marked "Degree of side cut of hip or valley," also of jack. To secure the angle of side cut it is only necessary to solve by simple trigonometric formulæ, the trianglea b cand thena′ b′ c′ofFig. 62.The angleAis the angle of inclination of hip or jack or valley, and will be found for each inch of rise in the tables; the manner of determining the same having been described. The sidebisA-BofFig. 74, the value of which is also easily determined once the principles ofFig. 74aremastered. With this data the student may find the angleA'ofFig. 62as given in the tables.

Rafter lengths are determined as previously described in connection with framing with the steel square.

Fig. 81. Side Cut Fig. 82. End Cut

44. Translating Framing Problems from Protractor to Framing Square and Vice Versa.—Frequently it is desirable to translate framing problems from degrees to numbers to be used upon the steel square, and vice versa. To change from degree framing to steel square framing it is only necessary to remember that the numbers to use on the square must be numbers such that their ratio one to the other shall give a tangent value equal to that given in the Table of Natural Functions, Appendix II, for the angle under consideration.

Example:Given: Angle of inclination of common rafter or of roof = 30 degrees.Find the numbers to take on the square to frame seat and plumb cuts.Solution:Tan 30° = .577 (by Table, Appendix II).By agreement, run of common rafter takes 12" on one member of thesquare for constant of run.Therefore, rise must be 6.93", or 611/12", must be taken on the othermember of the square. (The base of the table is 1 so that for12" run we must have 12" x .5774.)

In a similar manner the number to be taken on the blade, when the inclination of any other common or jack rafter is given, may bedetermined. In the case of the hip or valley inclination, however, it must be remembered that 17" is to be taken on the tongue for the run in the square cornered house, with 13" in the octagon, which will necessitate multiplying the tangent value of the angle of inclination by 17 and 13 respectively to find the number to take on the blade when one or the other of these is taken on the tongue.

Example:Given: Side cut of jack rafter, or hip or valley = 38 degrees.Find the numbers to use on the framing square to lay off thissame angle.Solution:Tan 38° = .7813By agreement, we shall take 12" on the tongue.The number to be taken on the blade must be, therefore,12" x .7813 = 9⅜(In the case of the side cut, any number other than 12" might havebeen assumed on the tongue.)Example:Given: A hip rafter on a square roof which is framed with 17" on thetongue and 8" on the blade.To find the angle of inclination of the hip.Solution:8/17 = .4706The angle whose tan = .4706 is 25° 19′

45. Framing an Octagon Bay.—Whatever may be one's opinion as to the propriety of the octagon bay architecturally, its very common use makes it obligatory upon the builder to know how to properly frame it.

Referring to Figs.72and83-ait will be seen that the octagon bay is but a portion of a full octagon set up against the side of a house. From this it follows that the framing of the plate, sills, the laying out of plumb and seat cuts, and side or cheek cuts of jacks that rest against hips, etc., will be according to principles discussed in previous sections of the text. The chief thing which needs explanation is the matter of laying out side cuts for hips and jacks which are to rest against the side of the building.

Consider the bay as cut along the lineI-J,Fig. 83-a. To frame the hip nearest the building: (1) Determine the angle which would give the side or cheek cut of the rafter involved when the rafter has no pitch, that is, when it lies in the plane of the plate.Fig. 83-aindicates the manner of determining this angle, which will be found to be 22½°. If we wish to use a framing square instead of a framing tool we may readily translate the 22½°. We shall find that 12" taken on the tongue of the square will require 4.97" or 5" on the blade. (2) Place the square as inFig. 83-band scribe along the tongue. Remember that this gives the miter when the rafter lies in the plane of the plate. This would give the cut to be used where ceiling joist or floor joist of octagon bay are run parallel with the rafters. On small bays, joists are seldom run thus but are run parallel with other joists of the house. No special directions are needed for setting the protractor, or the combination tool, the manner of setting and placing is so obvious. (3) Once having this miter when the rafter lies in the plane of the plate, proceed as directed inSec. 35, (2)et seq.With one such rafter framed the mates may be laid out by transposition.

To frame the remaining hips ofFig. 83-aproceed 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 toolor 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 whenb′= 1.Whenb′= 12" cotA′= 4.97" or 5".

Fig. 84-a. Fig. 84-b.Framing Octagon Hip Intercepted at 67½°

(1) Place the square as inFig. 84-band 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 inSec. 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 itlies in the plane of the plate. It is 45 degrees. Tan 45 degrees = 1 whenb″= 1. Tan 45 degrees = 12" whenb″= 12". (2) Place the square as inFig. 85-band scribe along the tongue. (3) From this on, continue as inSec. 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, asb,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 linesE-DandF-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 astick 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.

To determine the length of the intercepted hip rafter overcof trianglea b cFig. 83-a, the value ofbmust first be secured.

This value represents common rafter run for hip rafter rising over sidec, and run of intercepted common rafter rising over sideb.

Length of intercepted hip rising over sidec=b xunit length of hip. (Table of lengths of octagon hips in terms of run of common rafter,Fig. 73.)

Length of intercepted common rafter or jack nearest house =b xunit length of common rafter. (Table of lengths of common rafter in terms of run of common rafter,Fig. 49.)

It must be remembered that these are theoretic lengths measured down the middle of the top edge of the rafters. In framing, suitable reductions must be made. For example, in framing the intercepted common rafter nearest the building, the rafter must be set over one-half its thickness that it may be nailed against the side of the building. The allowance necessary in order to do this may be secured by measuring the length from the long point instead of the middle of the side or cheek cut. Suitable reduction, too, must be made for the hip thickness in this case one-half the diagonal across the top edge of the hip, laid off at an angle of 22½°, or 5" and 12" on the square. This amount will be laid off straight back from the plumb cut,Sec. 35, (2)et seq.

Length of hip must be measured along the middle of the top of the rafter.

46. Framing a Roof of One Pitch to Another of Different Pitch.—Occasionally one must frame a roof of one pitch against a roof of another pitch. To determine the cut, if the combination tool is to be used, merely find the difference in degrees of the anglesof inclination of the two rafters, and apply the tool as indicated in the tables for making any other plumb cut, Figs.49,60and73. The seat cut will be determined by the rise and run of the shed rafter in the usual manner.

Fig. 86. Laying off Cut of Shed Rafter

Where the framing square is to be used, lay the square as in framing a plumb cut on the shed rafter, as atA,Fig. 86, taking on the tongue, the run, and on the blade the rise of the shed rafter. Next lay a second square as atB,Fig. 86, taking on the blade the rise and on the tongue the run of the rafter of the main roof, using the blade ofAas a reference edge. Blade ofBgives the cut.

47. Framing a Roof of Uneven Pitch.—Not infrequently a roof must be framed in which several pitches are involved. All of the principles necessary for framing such a roof have been developed. It remains for the student to make the applications to uneven pitches. It is advisable to prepare a framing plan as shown inFig. 87.From such a plan it may be seen that the seat and plumb cuts of common and jack rafters are determined in the usual manner, being different upon the different pitches, of course, but determined as for any given pitch. Lengths of common rafters will be determined for any pitch by the tables already made use of, the run being known or determined.

In selecting the numbers to use on the tongue and the blade of the square, in laying out seat and plumb cuts of hip or valley rafters of intersecting roofs of different pitches, any numbers may be usedproviding they have a ratio equal to that of the run and rise of the rafter being framed. Since the angle of intersection changes with every change of pitch, it is hardly worth while developing a constant to be used on the tongue in framing hip and valley rafters on irregular pitches.

Side or cheek cuts for valley, hip or jacks are determined according to principles developed inSec. 35.

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