View Full Version : Math problem for the simple minded ME!

Mr Fixit
05-11-2013, 06:11 PM
Hello Group,
Trying to build and arch for a gate in the yard out of 4x4 lumber and for the life of me I am just not getting it.:confused: Algebra was as far as I got in school so I'm looking for some help here.
I have a 38" diameter, 19" radius, and want to build a semi circle for the top of the oppening. I divided it into 8 sections and got 7.5" @ 22.5* for each piece but my math was off because I got a 25" circle when I cut the pieces. Where did I go wrong and what is the correct angle of each cut and the length of each piece. again I'm not the sharpest tool in the shed :D so I hope somebody here can help.

TX Mr fixit for the family
Chris :o

Paul Alciatore
05-11-2013, 06:32 PM
A 38" diameter semi circle (as in HALF circle) divided into 8 sections. 180 degrees / 8 = 22.5 degrees. So far, so good. Now if you are cutting the pieces to fit around that half circle, each end of each piece must be cut at 1/2 that amount or 11.25 degrees from square.

Since you are using 4" wide lumber (which is actually 3.5" wide), the semi circle will have both an inside diameter and an outside diameter. You have to keep them straight. I am going to assume you want a 38" inside diameter and the outside diameter will then be 2 x 3.5" bigger than that. 38 + 7 = 45". Your segments will also have different lengths on the ID and on the OD. I like to go with the longer ones so I will use the OD. Now, these are chords, not arcs so you can not just divide half the circumference by 8. The math looks like this:

Chord Length = 2 ( radius * sin( half angle) )

Chord Length = 2 ( 22.5 * sin( 11.25) )

Chord Length = 2 ( 22.5 * 0.195 )

Chord Length = 8.779 inches

Now, remember this is the outer chord length, corresponding to the 45" circle.

I did a quick CAD sketch to verify the accuracy.

Edited to correct two mistakes. Chord is spelled c-h-o-r-d, not as I originally spelled it four times. I was tired. In my defense, I did spell it correctly once or twice.

Second correction was to remove this statement, "The tops of the verticals that this "arc" rests on will also need to be cut at 11.25 degrees from square." That is not correct. The verticals would be cut square.

05-11-2013, 06:35 PM
May seem simple, but it's something I could get caught making mistakes on-

I would think that you'd use the formula for circumference to start with, keeping in mind that the width of the pieces you're building with will affect the 'effective' circumference. Then you have what I'm assuming to be 180 degrees of arc, giving a half circle- an assumption again. If that's so, then 6 30 degree joins would seem right, but I think 5 would be right- the beginning and end of the arc would not count when dividing the half circle-

something like that anyway-

There you go- Paul did the math and probably has it right.

Dr Stan
05-11-2013, 06:57 PM
I find myself calculating circular segments several times a year and have always referred to the formulas in Machinery's. Once upon a time I even wrote a short program in Basic so I could use a TRS-80 (Trash 80 to those of us of a "certain age") to run the calculations. However, I recently had to calculate another one, started for my Machinery's Handbook, gave myself a Marine Corp salute (that's a smack up against one's forehead) and looked on-line. Sure enough I found this site: http://planetcalc.com/1421/

So I ran your numbers through the calculator and had the same result as Paul. Chord length = 8.78

Mr Fixit
05-11-2013, 07:34 PM
Guys, you have come through again for me and maybe someone else here, or in the "CLOUD" as they call it.
Thank you very much, I can now go buy a new 4x4 and try again with this new knowlege.
Extra TX's to Paul for the calc. and the great explanation.

Mr. fixit for the family
Chris :D

05-11-2013, 09:26 PM
That is a problem that my free software can solve.

05-12-2013, 08:23 AM
Given that there are many ways to do this, I am curious why you did not take what appears to be the simplest path -

Make a cardboard template (via a string and pencil) and use that template to mark each mocked up 4x4?

05-12-2013, 09:10 AM
Make a cardboard template (via a string and pencil) and use that template to mark each mocked up 4x4?Luddites! There's one in every group.;)

05-12-2013, 11:40 AM
A bit tongue in cheek, this level of detail may not make much difference in carpentry, but it shows how careful you need to be for more critical applications.

Being rather fussy/picky/pedantic :p, there is a very small flaw in the calculation. The length of the 11.25 cut is 1/cos(11.25) times longer than the width of the timber (~2%), so the outside diameter required is actually 45.137", which in turn makes the long edge of the arch pieces 8.806".

Similarly to make the arch pieces fit the top of the columns precisely, without overlap, both the column tops and the lowest edge of the bottom arch pieces should be cut at 5.625. Both cuts are then 3.517" long, rather than 3.569" and 3.5". (n.b. This also makes the outer edges of these two pieces shorter at 8.454", as the inner edges are always 7.413".)


As I say, fussy...;)