Steel square
The steel square is a tool used in carpentry. Carpenters use various tools to lay out structures that are square (that is, built at accurately measured right angles), many of which are made of steel, but the name "steel square" refers to a specific long-armed square that has additional uses for measurement, especially of various angles. It consists of a long, wider arm and a shorter, narrower arm, which meet at an angle of 90 degrees (a right angle). Today the steel square is more commonly referred to as the framing square or carpenter's square, and  such squares are no longer invariably made of steel (as they were many decades ago); they can also be made of aluminum or polymers, which are light and resistant to rust. The longer wider arm is wide, and is called the blade; the shorter narrower arm, is wide, and is called the tongue. The square has many uses, including laying out common rafters, hip rafters and stairs. It has a diagonal scale, board foot scale and an octagonal scale. On the newer framing squares there are degree conversions for different pitches and fractional equivalents. Framing squares may also be used  as winding sticks. Blade and tongue. In traditional timber frame joinery, mortises and tenons were typically wide and from the edge of the timber when working with softwoods, giving rise to the width of the blade. Likewise, mortises and tenons were traditionally wide when working in hardwoods, explaining the width of the tongue. This allowed for quick layouts of mortise and tenon joints when working both hard and softwoods. Use. Calibration. A steel square is self-proving and self-calibrating in that you can lay out a perpendicular line, flip the square over, and determine the size and direction of  the error. The error can be corrected by opening or closing the angle with a center punch. Stair framing. Stairs usually consist of three components. They are the stringer, the tread and the riser. The stringer is the structural member that carries the load of the staircase, the tread is the horizontal part that is stepped on, and the riser board is the vertical part which runs the width of the structure. There are many types of stairs: open, closed, fully housed, winding, and so on, to mention a few of them. Laying out a staircase requires rudimentary math. There  are numerous building codes to which staircases must conform. In an open area the designer can incorporate a more desirable staircase. In a confined area this becomes more challenging. In most staircases there is one more rise than there are treads. Roof framing. There is a table of numbers on the face side of the steel square; this is called the rafter table. The rafter table allows the carpenter to make quick calculations based on the Pythagorean theorem. The table is organized by columns that correspond to various slopes of the roof. Each column describes a different roof inclination (pitch ) and contains the following information: Octagon scale. The octagon scale allows the user to inscribe an octagon inside a square, given the length of the side of the square. The markings indicate half the length of the octagon's sides, which can be set to a compass or divider. Arcs drawn from the midpoints of the square's sides will intersect the square at the vertices of the planned octagon. All that remains is to cut four triangular sections from the square. Diagonal scale. Knee bracing is a common feature in timber framing to prevent racking under lateral loads. The  diagonal scale is useful for determining the length of the a knee brace desired for a given distance from the joint between the post and beam. Calculators in roof framing. In addition to use the square tool, construction calculators are also used to verify and determine roofing calculations. Some are programmed to calculate all side cuts for hip, valley and jack regular rafters to be exactly 45° for all rafter pitches. The rafter table is expressed in inches, and the higher the numerical value of the pitch, the greater the difference between side cut angles within a given pitch . Only a level roof, or a 0 pitch will require a 45° angle side cut (cheek cut) for hip and jack rafters. Side cut hip/valley rafter table. If a right triangle has two angles that equal 45° then the two sides are equidistant. The rafter is the hypotenuse and the legs or catheti of the triangle are the top wall plates of the structure. The side cut is located at the intersection of the given pitch column and the side cut of the hip/valley row. The regular hip/valley rafter runs at a 45° angle  to the main roof and the unit of measurement is 16.97 inches of run. Regular hip/valley and jack rafters have different bevel angles within any given pitch and the angle decreases as the pitch increases. The side cut of the hip/valley rafter = (Tangent)(12) = side cut in inches. The side cuts in the rafter table are all in a base 12. The arc tan can be determined from any given pitch. Most power tools and angle measuring devices use 90° as 0° in construction. The complementary angles of the arc tan are used with tools  like the speed square. Side cut of jack rafters. The side cut is located at the intersection of the side cut of jack rafters row and the pitch column on the Steel square. There is a row for the difference in length of jacks, 16 and 24 inch centers on the blade. The tangents are directly proportional for both centers. The tangent is in a base 12. The tangent x 12 = side cut of jack rafters. This corresponds to the side cut on the Steel square. The complementary angles of the arc tan are used on most angle measuring  devices in construction. The tangent of hip, valley, and jack rafters are less than 1.00 in all pitches above 0°. An eighteen pitch has a side cut angle of 29.07° and a two pitch has a side cut angle of 44.56° for jack rafters. This is a variation of 15.5° between pitches. Side cut angles versus pitch This is a reference table for side cuts versus pitch. (only valid for 90 degrees eave angle) : Pitch expressed in rise units / run units Pitch 18/12 ==> 60,86 deg Pitch 17/12 ==> 60,10 deg Pitch  16/12 ==> 59,07 deg Pitch 15/12 ==> 57,99 deg Pitch 14/12 ==>  56,94 deg Pitch 13/12 ==>  55,88 deg Pitch 12/12 ==>  54,69 deg Pitch 11/12 ==>  53,49 deg Pitch 10/12 ==>  52,54 deg Pitch 9/12 ==>  51,25 deg Pitch 8/12 ==>   50,19 deg Pitch 7/12 ==>   49,17 deg Pitch 6/12 ==>   48,15 deg Pitch 5/12 ==>   47,33 deg Pitch 4/12 ==>   46,54 deg Pitch 3/12 ==>   45,90 deg Pitch 2/12 = =>   45,22 deg Pitch 1/12 ==>   45,10 deg Pitch 0/12 ==>  45,00 deg Plumb cut of jack and common rafters. The plumb cut for jack and common rafters are the same angles. The level cut or seat cut is the complementary angle of the plumb cut. The notch formed at the intersection of the level and plumb cut Is commonly referred to as the bird's mouth . Plumb cut of hip/valley rafters. The plumb cut of the hip/valley rafter is expressed in the formula. The level cut is the complementary angle or 90 ° minus the arc tan. Irregular hip/valley rafters. The only Framing Square that has tables for unequal pitched roofs is the Chappell Universal Square, (patent #7,958,645). There is also a comprehensive rafter table for 6 & 8 sided polygon roofs (first time ever on a framing square). The traditional steel square's rafter table (patented April 23,1901) is limited in that it does not have tables that allow for work with unequal pitched roofs. Irregular hip/valley rafters are characterized by plan angles that are not equal or 45°. The top plates can be 90° at the outside  corners or various other angles. There are numerous irregular h/v roof plans. Carpenter's square. In carpentry, a square is a guide for establishing right angles (90° angles) or mitre angles, usually made of metal. There are various types of square, such as speed squares, try squares and combination squares.