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 plate
for 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".
 Fig. 71-a. |  Fig. 71-b. |
| Laying out Miters | |
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 tangent
of 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 manner of 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. [68] and [72] it will be seen that common rafters have for their runs the apothem of the polygon made by the plate, represented in [Fig. 68] by the lines b, b’, b″. The run of the hip is represented by the line c, 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 in [Sec. 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 formula c² = 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.