Roof Slope vs Roof Angle

[kinelbor]

Came across a strange problem in my Carpentry class the other day. The shop assistant is building a gazebo, the plan says to use a 6/12 roof. Well he figured that would be a 22.5 degree angle and built it by that. He had me layout one of the rafters and I used the framing square on 6 and 12. Its not the same angle! Few of us thought about it and cant figure out why. A 12/12 is a 45 degree angle. I did the math out, tan= opposite over adjecent, so inverse tangent of 6/12 is 26.56 degrees. Inverse tan of 12/12 is 45. Can someone explain how this is possible?

[Jim Becker]

C'mon Nik, draw a few pictures and think this through. You think a 24/12 roof would be a 90 degree angle?

Here is an exercise that should make it more clear. Make a small table of values for the tangent of an angle by measuring a drawing:

You will need a ruler and a protractor. Your angles and measurements will not need to be very precise to get the idea.

Take a piece of paper and pick a point near the lower left corner. Draw a base line across, whatever length works easily with the protractor. Draw one line up from each end of the base line, perpendicular to the base line.

Use the protractor to draw lines from the original point at several angles to the base until they cross the line on the right. Measure the length of the segment of the line on the right between the base line and the point the angled line crosses it. Divide that by the length of the base at several angles (same as roof slope) to get the tangent.

After you have made a small table of these numbers, try drawing a graph of angle vs. tangent with the x-axis 0 to 90 degrees and the tangent on the y-axis.

I suggest using angles of 0, 15, 30, 45, 60, 75 and 90. You might also want to do 22.5 since it was part of your original question.

[kinelbor]

Of course Never thought to think about slopes above 12/12 and that they can never reach 90 degrees. Now it makes sense! Thanks Jim. Now I have to figure out a way to explain to the shop assistant that all the angles that he cut at 22.5 degrees are incorrect.

[Jim Becker]

Tell the man that is in charge of the shop and let him explain it to the assistant?

[George Willer]

Well he figured that would be a 22.5 degree angle and built it by that. He had me layout one of the rafters and I used the framing square on 6 and 12. Its not the same angle!

Nik,

It's been many years since I framed a roof but in the past I've built dozens of houses with 5/12 pitch. It isn't exact, but very close to 22 1/2 degrees.

[Eugene]

http://www.roofvents.com/pitchdiagram.html

[George Willer]

http://www.roofvents.com/pitchdiagram.html

Somebody published that chart without checking it. Their figures can be safely ignored. 5/12 is correctly described as the steepest that can be easily walked on but their angle is WAY off. I hope noone uses their angles to frame with. Their quoted angle for 5/12 is 19 degrees when it is actually 22.62 degrees from horizontal.