Example:
Find the value of the tangent for an octagonal plate.
Solution:
Angle A′ of [Fig. 68] = 22½°
(1/16 of the sum of all the angles about a point)
Tan 22½° = .4143
Tables are builded with 1 as a base. In roof framing 1" or 12" is
taken 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 buildings
of any number of sides.
In [Fig. 69] is 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. The reverse 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-b] illustrates 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 in [Sec. 35].
Third, this tangent value is needed in determining the amount of backing to be given hip rafters. This is discussed in [Sec. 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°
 Fig. 70-a. |  Fig. 70-b. |
| Griffith's Roof Framing Tables | |