[Illustration: Fig. 12. Graphic Illustration of Proportional Dimensions of Treads and Risers.]
In the right-angle triangle _A B C_, make _A B_ equal to 24 inches, and _B C_ equal to 11 inches—the standard proportion. Now, to find the riser corresponding to a given width of tread, from _B_, set off on _A B_ the width of the tread, as _B D_; and from _D_, erect a perpendicular _D E_, meeting the hypotenuse in _E_; then _D E_ is the height of the riser; and if we join _B_ and _E_, the angle _D B E_ is the angle of inclination, showing the slope of the ascent. In like manner, where _B F_ is the width of the tread, _F G_ is the riser, and _B G_ the slope of the stair. A width of tread _B H_ gives a riser of the height of _H K_; and a width of tread equal to _B L_ gives a riser equal to _L M_.
In the opinion of many builders, however, a better scheme of proportions for treads and risers is obtained by the following method:
Set down two sets of numbers, each in arithmetical progression—the first set showing widths of tread, increasing by inches; the other showing heights of riser, decreasing by half-inches.
TREADS, INCHES RISERS, INCHES 5 9 6 8½ 7 8 8 7½ 9 7 10 6½ 11 6 12 5½ 13 5 14 4½ 15 4 16 3½ 17 3 18 2½
It will readily be seen that each pair of treads and risers thus obtained is suitably proportioned as to dimensions.
It is seldom, however, that the proportions of treads and risers are entirely a matter of choice. The space allotted to the stairs usually determines this proportion; but the above will be found a useful standard, to which it is desirable to approximate.
In the better class of buildings, the number of steps is considered in the plan, which it is the business of the Architect to arrange; and in such cases, the height of the story rod is simply divided into the number required.
=Pitch-Board.= It will now be in order to describe a pitch-board and the manner of using it; no stairs can be properly built without the use of a pitch-board in some form or other. Properly speaking, a pitch-board, as already explained, is a thin piece of material, generally pine or sheet metal, and is a right-angled triangle in outline. One of its sides is made the exact height of the rise; at right angles with this line of rise, the exact width of the tread is measured off; and the material is cut along the hypotenuse of the right-angled triangle thus formed.