 WORK IN PROGRESS carpentermath to numbers page and more detail at ROOFMATH Similar means that all angles are the same and that all sides are in direct proportion to each other. Rule number one keep it simple reduce all runs to the number "1" for example a rise of 10 and a run of 12 reduces to 10/12 this = run of "1" and the rise becomes a percentage of "1" of 83% or .833 now for all triangles similar to this the run is a multiple of 1 and the rise is a multiple of.833 now how is this convenient...if cutting a plumb cut on a 2x4 you know the run is the 3.5" the rise is.833 times 3.5" = 2.91"  what about cross blocking on irregular spaced joists ..height of joist is rise the space between joist is the run Carpenters math is based on geometry and trig it uses the concept of similar triangles, (NOTE ALL squares and circles are similar)... were all similar shapes are directly in ratio and proportion to each other.. we can determine the sizes of materials and the various shapes and angles needed to build solving ratio and proportions of similar triangles are based on cross multiplication you must know three of the numbers to find the fourth for example two similar triangles the first triangle having a rise of 4 and a run of 6 the second triangle has a run of 9 the rise would be 4/6*9=6 Note make it simple base every thing on the number1 example: 4/6 =.66 that means for this similar triangle a run of 1 = a rise of .66 there fore for every triangle similar to this triangle for every 1 of run there is .66 of rise Also note the .66 = tan and the tan of .66 = your angle of rise Rule of thumb if in field with out a calculator half of tan -5 is very close to your angle at 45 degrees and you subtract less for smaller angles until 30 were it is close to half of tan  The circle above and all the sizes of each line is a result of the simple triangle laid out in green with the sides labeled rise, rake and run... they are all based on similar triangles and are in direct ratio and proportion to each other... this particular diagram also depicts the octagon and square found with in each circle NOTE for every right angle triangle ...rise squared plus run squared divided by 2 rises = radius of the that triangles circle ...rise squared plus run squared = rake squared ....rake squared minus run squared= rise squared... the square root of the rise squared = the rise of the triangle and the height of the arc (below)  Known numbers... the area of the square around the circle is the diameter squared the area of the circle is .785 times the area of the square... the square formed with in the circle has an area of.633 of the circle the rake of a 1/1 triangle (rise1 / run 1 )is 1.41   The overhang is a similar triangle to the major roof thus the run of of the overhang gives you the rise and the rake...remember run squared plus rise squared = rake squared and the square root of this gives the rake of the roof also known as the line length wp1 stand for work point 1 the ridge cut wp2 is for work point 2 the birdsmouth wp3 is for work point 3 the tail cut  The green indicates the true barn or gambrel roof Note how it fits into the octagon with in the circle So the rafters are all equal lengths The top chord would be the collar tie (marked with red) The rise of the bottom rafter becomes a pony wall (marked in red) The bottom and top rafters have opposite rises and runs The rises indicate the plumb cuts The runs are the birdsmouth cuts (1/3 depth of stock) HIP ROOF  The runs of the common rafter times 1.412 = run of hip rafter The hip jack rafters the run = the distance from corner of building so 16" centres would give runs of 16..32..48.. etc. when cutting you can use the common difference for each rafter in progression (each rafter grows exactly the same amount with each cut)    Rule the tangent angle to it's chord is 1/2 the angle of the two radius that form the chord  The runs of the common rafter times 1.412 = run of valley rafter The valley jack rafters the run = the distance from corner of the ridges so 16" centres would give runs of 16..32..48.. etc. when cutting you can use the common difference for each rafter in progression (each rafter grows exactly the same amount with each cut) 
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