The framing square is one of the most frequently used Builder tools. The problems it can solve are so many and varied that books have been written on the square alone. Only a few of the more common uses of the square can be presented here. For a more detailed discussion of the various uses of the framing square in solving construction problems, you are encouraged to obtain and study one of the many excellent books on the square.
The framing square (fig. 2-5, view A) consists of a wide, long member called the blade and a narrow, short member called the tongue. The blade and tongue form a right angle. The face of the square is the side one sees when the square is held with the blade in the left hand, the tongue in the right hand, and the heel pointed away from the body. The manufacturers name is usually stamped on the face. The blade is 24 inches long and 2 inches wide. The tongue varies from 14 to 18 inches long and is 1 1/2 inches wide, measured from the outer corner, where the blade and the tongue meet. This corner is called the heel of the square.
The outer and inner edges of the tongue and the blade, on both face and back, are graduated in inches. Note how inches are subdivided in the scale on the back of the square. In the scales on the face, the inch is subdivided in the regular units of carpenters measure (1/8 or 1/16 inch). On the back of the square, the outer edge of the blade and tongue is graduated in inches and twelfths of inches. The inner edge of the tongue is graduated in inches and tenths of inches. The inner edge of the blade is graduated in inches and thirty-seconds of inches on most squares. Common uses of the twelfths scale on the back of the framing square will be described later. The tenths scale is not normally used in roof framing.
Figure 2-5.Framing square: A. Nomenclature; B. Problem solving.
SOLVING BASIC PROBLEMS WITH THE FRAMING SQUARE
The framing square is used most frequently to find the length of the hypotenuse (longest side) of a right triangle when the lengths of the other two sides are known. This is the basic problem involved in determining the length of a roof rafter, a brace, or any other member that forms the hypotenuse of an actual or imaginary right triangle.
Figure 2-5, view B, shows you how the framing square is used to determine the length of the hypotenuse of a right triangle with the other sides each 12 inches long. Place a true straightedge on a board and set the square on the board so as to bring the 12-inch mark on the tongue and the blade even with the edge of the board. Draw the pencil marks as shown. The distance between these marks, measured along the edge of the board, is the length of the hypotenuse of a right triangle with the other sides each 12 inches long. You will find that the distance, called the bridge measure, measures just under 17 inches16.97 inches, as shown in the figure. For most practical Builder purposes, though, round 16.97 inches to 17 inches.
Solving for Unit and Total Run and Rise
In figure 2-5, the problem could be solved by a single set (called a cut) of the framing square. This was due to the dimensions of the triangle in question lying within the dimensions of the square. Suppose, though, you are trying to find the length of the hypotenuse of a right triangle with the two known sides each being 48 inches long. Assume the member whose length you are trying to determine is the brace shown in figure 2-6. The total run of this brace is 48 inches, and the total rise is also 48 inches.
Figure 2-6."Stepping off" with a framing square.
To figure the length of the brace, you first reduce the triangle in question to a similar triangle within the dimensions of the framing square. The length of the vertical side of this triangle is called unit of rise, and the length of the horizontal side is called the unit of run. By a general custom of the trade, unit of run is always taken as 12 inches and measured on the tongue of the framing square.
Now, if the total run is 48 inches, the total rise is 48 inches, and the unit of run is 12 inches, what is the unit of rise? Well, since the sides of similar triangles are proportional, the unit of rise must be the value of x in the proportional equation 48:48::12:x. In this case, the unit of rise is obviously 12 inches.
To get the length of the brace, set the framing square to the unit of run (12 inches) on the tongue and to the unit of rise (also 12 inches) on the blade, as shown in figure 2-6. Then, "step off" this cut as many times as the unit of run goes into the total run. In this case, 48/12, or 4 times, as shown in the figure.
In this problem, the total run and total rise were the same, from which it followed that the unit of run and unit of rise were also the same. Suppose now that you want to know the length of a brace with a total run of 60 inches and a total rise of 72 inches, as in figure 2-7. Since the unit of run is 12 inches, the unit of rise must be the value of x in the proportional equation 60:72::12.x. That is, the proportion 60:72 is the same as the proportion 12:x. Working this out, you find the unit of rise is 14.4 inches. For practical purposes, you can round this to 14 3/8.
To lay out the full length of the brace, set the square to the unit of rise (14 3/8 inches) and the unit of run (12 inches), as shown in figure 2-7. Then, step off this cut as many times as the unit of run goes into the total run (60/12, or 5 times).
Determining Line Length
If you do not go through the stepping-off procedure, you can figure the total length of the member in question by first determining the bridge measure. The bridge measure is the length of the hypotenuse of a right triangle with the other sides equal to the unit of run and unit of rise. Take the situation shown above in figure 2-7. The unit of run here is 12 inches and the unit of rise is 14 3/8 inches. Set the square to this cut, as shown in figure 2-8, and mark the edges of the board as shown. If you measure the distance between the marks, you will find it is 18 3/4 inches. Bridge measure can also be found by using the Pythagorean theorem: a2 + b2 = c2. Here, the unit of rise is the altitude (a), the unit or run is the base (b), and the hypotenuse (c) is the bridge measure.
Figure 2-7."Stepping off" with a square when the unit of runa nd unit of rise are different.
Figure 2-8.-Unit length.
To get the total length of the member, you simply multiply the bridge measure in inches by the total run in feet. Since that is 5, the total length of the member is 18 3/4 x 5, or 93 3/4 inches. Actually, the length of the hypotenuse of a right triangle with the other sides 60 and 72 inches long is slightly more than 93.72 inches, but 93 3/4 inches is close enough for practical purposes.
Once you know the total length of the member, just measure it off and make the end cuts. To make these cuts at the proper angles, set the square to the unit of run on the tongue and the unit of rise on the blade and draw a line for the cut along the blade (lower end cut) or the tongue (upper end cut).
A framing square contains four scales: tenths, twelfths, hundredths, and octagon. All are found on the face or along the edges of the square. As we mentioned earlier, the tenths scale is not used in roof framing.
The graduations in inches, located on the back of the square along the outer edges of the blade and tongue, are called the twelfths scale. The chief purpose of the twelfths scale is to provide various shortcuts in problem solving graduated in inches and twelfths of inches. Dimensions in feet and inches can be reduced to 1/12th by simply allowing each graduation on the twelfths scale to represent 1 inch; for example, 2 6/12 inches on the twelfths scale may be taken to represent 2 feet 6 inches. A few examples will show you how the twelfths scale is used.
Suppose you want to know the total length of a rafter with a total run of 10 feet and a total rise of 6 feet 5 inches. Set the square on a board with the twelfths scale on the blade at 10 inches and the twelfths scale on the tongue at 6 5/12 inches and make the usual marks. If you measure the distance between the marks, you will find it is 11 11/12 inches. The total length of the rafter is 11 feet 11 inches.
Suppose now that you know the unit of run, unit of rise, and total run of a rafter, and you want to find the total rise and the total length. Use the unit of run (12 inches) and unit of rise (8 inches), and total run of 8 feet 9 inches. Set the square to the unit of rise on the tongue and unit of run on the blade (fig. 2-9, top view). Then, slide the square to the right until the 8 9/12-inch mark on the blade (representing the total run of 8 feet 9 inches) comes even with the edge of the board, as shown in the second view. The figure of 5 10/12 inches, now indicated on the tongue, is one-twelfth of the total rise. The total rise is, therefore, 5 feet 10 inches. The distance between pencil marks (10 7/12 inches) drawn along the tongue and the blade is one-twelfth of the total length. The total length is, therefore, 10 feet 7 inches.
Figure 2-9.-Finding total rise and length when unit of run, unit of rise, and total run are known.
The twelfths scale may also be used to determine dimensions by inspection for proportional reductions or enlargements. Suppose you have a panel 10 feet 9 inches long by 7 feet wide. You want to cut a panel 7 feet long with the same proportions. Set the square, as shown in figure 2-9, but with the blade at 10 9/12 inches and the tongue at 7 inches. Then slide the blade to 7 inches and read the figure indicted on the tongue, which will be 4 7/12 inches if done correctly. The smaller panel should then be 4 feet 7 inches wide.
The hundredths scale is on the back of the tongue, in the comer of the square, near the brace table. This scale is called the hundredths scale because 1 inch is divided into 100 parts. The longer lines indicate 25 hundredths, whereas the next shorter lines indicate 5 hundredths, and so forth. By using dividers, you can easily obtain a fraction of an inch.
The inch is graduated in sixteenths and located below the hundredths scale. Therefore, the conversion from hundredths to sixteenths can be made at a glance without the use of dividers. This can be a great help when determining rafter lengths, using the figures of the rafter tables where hundredths are given.
The octagon scale (sometimes called the eight-square scale) is located in the middle of the face of the tongue. The octagon scale is used to lay out an octagon (eight-sided figure) in a square of given even-inch dimensions.
Lets say you want to cut an 8-inch octagonal piece for a stair newel. First, square the stock to 8 by 8 inches and smooth the end section. Then, draw crossed center lines on the end section, as shown in figure 2-10. Next, set a pair of dividers to the distance from the first to the eighth dot on the octagon scale, and layoff this distance on either side of the centerline on the four slanting sides of the octagon. This distance equals one-half the length of a side of the octagon.
Figure 2-10.Using the octagon square.
When you use the octagon scale, set one leg of the dividers on the first dot and the other leg on the dot whose number corresponds to the width in inches of the square from which you are cutting the piece.
FRAMING SQUARE TABLES
There are three tables on the framing square: the unit length rafter table, located on the face of the blade; the brace table, located on the back of the tongue; and the Essex board measure table, located on the back of the blade. Before you can use the unit length rafter table, you must be familiar with the different types of rafters and with the methods of framing them. The use of the unit length rafter table is described later in this chapter. The other two tables are discussed below.
The brace table sets forth a series of equal runs and rises for every three-units interval from 24/24 to 60/60, together with the brace length, or length of the hypotenuse, for each given run and rise. The table can be used to determine, by inspection, the length of the hypotenuse of a right triangle with the equal shorter sides of any length given in the table. For example, in the segment of the brace table shown in figure 2-11, you can see that the length of the hypotenuse of a right triangle with two sides 24 units long is 33.94 units; with two sides 27 units long, 38.18 units; two sides 30 units long, 42.43 units; and so on.
Figure 2-11.-Brace table.
By applying simple arithmetic, you can use the brace table to determine the hypotenuse of a right triangle with equal sides of practically any even-unit length. Suppose you want to know the length of the hypotenuse of a right triangle with two sides 8 inches long. The brace table shows that a right triangle with two sides 24 inches long has a hypotenuse of 33.94 inches. Since 8 amounts to 24/3, a right triangle with two shorter sides each 8 inches long must have a hypotenuse of 33.94 ÷3, or approximately 11.31 inches.
Suppose you want to find the length of the hypotenuse of a right triangle with two sides 40 inches each. The sides of similar triangles are proportional, and any right triangle with two equal sides is similar to any other right triangle with two equal sides. The brace table shows that a right triangle with the two shorter sides being 30 inches long has a hypotenuse of 42.43 inches. The length of the hypotenuse of a right triangle with the two shorter sides being 40 inches long must be the value of x in the proportional equation 30.42.43::40:x, or about 56.57 inches.
Notice that the last item in the brace table (the one farthest to the right in fig. 2-11) gives you the hypotenuse of a right triangle with the other proportions 18:24:30. These proportions are those of the most common type of unequal-sided right triangle, with sides in the proportions of 3:4:5.
The primary use of the Essex board measure table is for estimating the board feet in lumber of known dimensions. The inch graduations (fig. 2-12, view A) above the table (1, 2, 3, 4, and so on) represent the width in inches of the piece to be measured. The figures under the 12-inch graduation (8, 9, 10, 11, 13, 14, and 15, arranged in columns) represent lengths in feet. The figure 12 itself represents a 12-foot length. The column headed by the figure 12 is the starting point for all calculations.
Figure 2-12.-Segment of Essex board measure table.
To use the table, scan down the figure 12 column to the figure that represents the length of the piece of lumber in feet. Then go horizontally to the figure directly below the inch mark that corresponds to the width of the stock in inches. The figure you find will be the number of board feet and twelfths of board feet in a 1-inch-thick board.
Lets take an example. Suppose you want to figure the board measure of a piece of lumber 10 feet long by 10 inches wide by 1 inch thick. Scan down the column (fig. 2-12, view B) headed by the 12-inch graduation to 10, and then go horizontally to the left to the figure directly below the 10-inch graduation. You will find the figure to be 84, or 8 4/12 board feet. For easier calculating purposes, you can convert 8 4/12 to a decimal (8.33).
To calculate the cost of this piece of lumber, multiply the cost per board foot by the total number of board feet. For example, a 1 by 10 costs $1.15 per board foot. Multiply the cost per board foot ($1. 15) by the number of board feet (8.33). This calculation is as follows:
What do you do if the piece is more than 1 inch thick? All you have to do is multiply the result obtained for a 1-inch-thick piece by the actual thickness of the piece in inches. For example, if the board described in the preceding paragraph were 5 inches thick instead of 1 inch thick, you would follow the procedure described and then multiply the result by 5.
The board measure scale can be read only for lumber from 8 to 15 feet in length. If your piece is longer than 15 feet, you can proceed in one of two ways. If the length of the piece is evenly divisible by one of the lengths in the table, you can read for that length and multiply the result by the number required to equal the piece you are figuring. Suppose you want to find the number of board feet in a piece 33 feet long by 7 inches wide by 1 inch thick. Since 33 is evenly divisible by 11, scan down the 12-inch column to 11 and then go left to the 7-inch column. The figure given there (which is 65/12, or 6.42 bd. ft.) is one-third of the total board feet. The total number of board feet is 6 5/12 (or 6.42) x 3, or 19 3/12 (or 19.26) board feet.
If the length of the piece is not evenly divisible by one of the tabulated lengths, you can divide it into two tabulated lengths, read the table for these two, and add the results together. For example, suppose you want to find the board measure of a piece 25 feet long by 10 inches wide by 1 inch thick. This length can be divided into 10 feet and 15 feet. The table shows that the 10-foot length contains 8 4/12 (8.33) board feet and the 15-foot length contains 12 6/12 (12.5) board feet. The total length then contains 8 4/12 (8.33) plus 12 6/12 (12.5), or 20 10/12 (20.83) board feet.