The square root of 2, (1.4142), is a magic number when it comes to hip roofs.
To find the horizontal length of a 45 degree hip rafter, multiply the setback by the square root of 2.
i.e. setback = 8' x square root 2 , (1.4142) = 11.3137' or 11-3-12
This works for the pitch of the hip rafter also, divide the main pitch by the square root of 2.
i.e. pitch =  6 / square root 2 = 4.2426. Pitch of the hip rafter is 4.2426/12.
Many framers use the pitch on 17 for the hip rafter, which is very close.
Lets prove it, we use pitches on 12, so multiply 12 by the square root of 2, which is 12 x 1.4142 = 16.9706, close enough for a framing square.

On a fink truss, if the panel points are divided equally from the inside of the scarf cut, the long web is twice as long as the short web.

To quickly calculate the short web center line length for 4/12 pitch, multiply the inside of scarf to inside of scarf length in feet by 1/8.5.
i.e. overall span = 24', scarf length with ¼" heel cut = .8125 feet. 24-.8125-.8125 = 22.375.
Multiply 22.375 x 1/8.5 = 2.6324 feet or 2-7-9
The magic number for 5/12 is 1/7.5 and the scarf is .65 feet for a ¼" heel cut.
Not too much use for this today, but back when truss plate suppliers charged for time used on their computers, we used this method for hand calculating web lengths on finks.

Calculating bastard hip set back dimensions may seem to be a nightmare, but is actually very easy to calculate.
Lets look at a 24 foot 6/12 main roof with a 10/12 hip end pitch and a 24" overhang.
Lets make the eave height on the 10/12 hip end, match the eave height of the 6/12 roof.
Sound complicated yet?
Doing  some simple math you will find the hip ridge setback to be 6.4 feet or 6-4-13 from the outside of the hip end wall.
To calculate this: Calculate the height of the 6/12 roof from the top of the end of the overhang. (12'+2') x 6 = 84 inches.
Divide the 84" by 10, which is the hip end pitch, to get 8.4 feet, subtract the 2 foot overhang to get 6.4 feet.
Very simple!
Now to calculate the heel height of the 10/12 roof to make the eave height match, use our charts.

Now for a little more complex.
How do you calculate where the peak is located on a dual pitch truss.
Lets look at a 24' dual pitch truss. 6/12 one side 4/12 the other side, with standard heel heights using a ¼" heel cut.

Step 1: Subtract the higher pitch heel height, in inches, from the other heel height. 6/12 height = 4.1631 inches. 4/12 heel height = 3.9393 inches, 3.9393-4.1631 =  -.2238 inches, note the negative number.

Step 2: Add the two pitches together, which is 10.

Step 3: Multiply the span by the lower pitch, 24 x 4 = 96

Step 4: Add the result in step 1 to step 3 then divide the result by step 2.  This gives you the distance to the peak on the 6/12 end.
(-.2238+96)/10 = 9.5776 feet or 9-6-15

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