Any Mathematicians around

[Redoverfarm]

I have a neighbor (close to Dogtrot) who is erecting a Quonset Hut as a residence.  The diameter is 27' and the circumference is 38'10".  He is currently in the erection stage sitting the ends on a 4' poured knee wall which will give him room enough for a small loft.  He is trying to figure out what the angle degree would be so that he can later frame partitions from one side over to the other side.      Anyone have any idea?

[flyingvan]

I'm trying to picture what your numbers mean--

   I'm guessing you're describing a circle with a line through it but not through its center, where the line (chord) is 27' and the circle part (arc length) is 38'10".  Is that correct?  Then, you want to know what the angle is in that curved part where a loft floor/partition wall plate would touch, leaving headroom for the bottom floor?
   If I'm looking at that correctly, the height not counting the stem wall is 11.63' (so 15.63' with the stemwall)...Let's say the ceiling below the loft plus the joists and floor for the loft add up to 9' even, giving a respectable 6 2/3' headroom at the high point...That makes the chord for the loft floor 23.4 feet across.  The angle where it hits the curved wall/roof is 'about' 30 degrees----more obtuse at the bottom, more acute at top.  Curves are funny that way.
   So if your friend needs to figure out now exactly where the walls will sit on the ground to rise up to this point, it's 1'9-1/2" in from the stemwall (that's the nearest point so the width of the wall goes towards the center of the room from there)

[Redoverfarm]

flyingvan I guess you could describe the metal structure as a Semi-circle.  The complete metal on top of the stem wall is suppose to be a half circle.  At the bottom of that the distance is 27' from one side to the other.

He is planning on cutting 2X4's in small segments (24" or so) to make curve from the top of the stemwall to the other side of the building to the opposite stemwall top.  I assume he will use straight 4' length stud to rise from the floor to the top of the stemwall where he will start that gradual curve. 

Don't quote me but I think he said that in the center the height would be around 13'9" to the lowest portion of the step in the metal. 

As for the loft I am not sure exactly where or how he is going to design it but he did indicate the floor will go from side to side. 

[flyingvan]

what does the 38' 10" circumference refer to?

If you could draw a picture with any measurements you have and a question mark where you need an angle or dimension it might be easier.

If it's truly a full 180 degree semicircle it makes things pretty easy.  The height will be half the diameter.

Seems to me the easiest way to support a loft inside a curved structure wouldn't be parallel to the flat end wall but perpendicular.  Have a beam inside or against that end wall then another spanning wherever you want the loft to end, then run your joists over them (or between with joist hangers to maximize headroom)

[Redoverfarm]

That 38'10" is the distance from the top of the stem wall (bottom of the metal) from one side to the other following the inside of the metal surface.

[Don_P]

Circumference is pi x diameter. divide that in half for circumference of a true half circle. Like flyingvan mentioned, it sounds like it might be a segment of a circle rather than a true semicircle, I'm coming up with 42.39' as the half circle circumference. If it is a true 180 degrees, then divide the 180 by the number of pieces desired to make up the arch to give the angle at each intersection. Divide that angle in half to bisect, miter, the ends of each stick.

What most people don't realize is that arches also thrust horizontally under load, just like a roof. He should check with the supplier to see if anything needs to be done to stabilize or restrain the stemwalls or if it is rigid enough not to thrust.

[Medeek]

With this sort of thing draw it up in Draftsight or AutoCAD and save yourself the headache of the math.  Some things are easier done by not doing them.

[flyingvan]

   I thought of a more satisfying answer for your friend, I think.  If the numbers you gave are accurate (27' chord and 38' 10" segment arc) if that metal part continued on to form a complete circle it would be 85'9" around and 27'3" across.
   So your friend wants to know the angles if he were to form a series of angles where the joints just touch the metal of the shed.  You threw out the '24" or so' length.

   A better approach is to decide how many of these lengths he wants to end up with.

If he went with 90 degree angles, he'd have three boards---the one across the top would be 19'3".

45 degree angles would be 5 boards, each 10' 5.5".  Still too big.

SO.  If he wants each board exactly 2' long at its longest, it'll be 21 and 1/2 boards in a semicircle, each angle 8.37 degrees (the cuts would be half that of course).  The regular polygon inscribed in the circle is a 43 sided monster called a tetracontakaitrigon.  Pretty cumbersome.

Instead, you gave some wiggle room with the 'about' 2 feet.  If we work with a 36 sided regular polygon, (triacontakaihexagon)  each board is a pretty convenient length of 28 and 3/4", with each angle being 10 degrees (cuts on each end of each board is 5 degrees.  There's even a line on the chopsaw for that!)

[Redoverfarm]

Flyingvan thanks for your reply.  This is closer to what he was looking for.  Again thanks for your time in researching this.