Framer's Roof Calculation

by Frederick Hoehn, copyright 2012.

One branch of Carpentry is Framing.  A friend of mine who is a Carpenter and a Framer asked if there’s a way to calculate the length of the slant two by fours for a roof if you have the vertical and the horizontal lengths.

The slant two by fours go on the building after the vertical and horizontal pieces are already there.  But it’s more convenient if you can cut those slant roof pieces with your saw on the ground rather than up on top of the building.

The Pythagorean Theorem will give you the length.  It’s for use with right triangles.  A right triangle has a right (or, ninety degree) angle. 

You can do it all in feet, or you can do it all in inches.  

Mathematicians know that there’s a three, four, five right triangle.  If the vertical is three feet, and the horizontal is four feet, then the slant length is five feet, in which case, no calculations needed.  

Let’s call the vertical V, the horizontal H, and the slant length S.  The formula is V multiplied by itself (squared) plus H squared equals S squared.

Testing that with our 3—4—5 right triangle, 3 squared is nine.  Four squared is sixteen.  Nine plus sixteen is twenty five.  Now let’s take the square root of twenty five.  What number when squared equals twenty five?  Five, of course.

But with other lengths, we can use the square root function on a hand calculator.

Before we start calculating, let’s make sure that there is zero in the calculator’s memory.  There’s a little Casio hand calculator on the table here.  Press memory recall (“RMC”) to see what’s in memory.  

If there’s anything other than zero in memory, I got rid of it by pressing “M-“, subtracting what was in the display from memory.  Press clear to get zero in the display, now again recall memory to find zero.

What if the vertical (V) is three and a half feet (3.5) and the horizontal (H) is twelve feet.  Then what would the slant length (S) be?
Press 3  .  5.  Then press the multiply key (“X”), then the Equals key, and you get 12.25 in the display, which is 3.5 squared. 

Now press “M+” to put that result in memory.  Now press clear to get zero in the display. 

Now press 1   2, then “X”, then Equals.  So we have 12 squared, which is 144. 
 
Now press “M+” to add it to what’s already in memory.

Now recall memory, and we have 156.25.  Now press the square root key, which looks like a V with a horizontal line attached to the upper right of the V, and you get the result of 12.5 feet for your slant piece.

But remember, most roofs have an overhang.  The Pythagorean Theorem only covers the length of the hypotenuse of the triangle.  Now add to that however much overhang the architectural plans call for.

Remember also that the theorem deals with lines on a piece of paper, but houses are built with lumber that has not only length, but also width and thickness.  So it would be wise to allow an extra inch or two the first few times of applying this method.  You can always cut a little more off, but it’s harder to put length back on.

There’s another method to calculate the slant length using Trigonometry.  Let’s use the same example of V=3.5 feet, and H=12 feet.  

This little Casio calculator doesn’t do trig functions, so we’ll use a Texas Instruments calculator.  Let’s call the angle between the horizontal and slant theta, a letter of the Greek alphabet.  We’ll just use T for theta.
The tangent of T is the ratio of V divided by H. 

We’ll get the angle of T using the arctan function on the calculator.  It’s the alternate function of the “TAN” key.  So, press “2nd” to get alternate function.  Then press “TAN”.  Then press 3  .  5, followed by the division key, followed by 1   2, then the right parenthesis key.  Then “Enter/=”.

The result is approximately 16.26 degrees, which we save in memory.  

Now the formula for our slant length is S=(H) x (secant of T).  There’s no secant key, but the secant is the reciprocal of the cosine.  

Clear the display.  Press “COS” for cosine, then “2nd”,  “STO” to recall the angle.  Then right parenthesis.  Then equals.  We see the cosine of our angle, T. 

Now, to get the reciprocal of the cosine, press the “x to the negative 1 power” key, followed by equals.  That gives us the secant of the angle.    

Then press “X” for multiply, followed by 1  2  (our H of 12 feet), followed by equals, and we get 12.5 feet for the S slant length.