I was given a job to build a frame with diagonal slats spanning the inside,
where the corner of the end slats fitted into the corner of the frame and the slats met equally throughout the frame, at a given angle.
It was easy doing it with three slats for this job, however, I thought there must be some kind of formula, that I could devise to be able to solve for the exact width of the slats, so as to save time and materials cutting and fitting trying t get an even array of angled slats.
So I drew up a scenario of a frame with a given number of slats, and began to geometrically break it down into singular shapes where I could write up equations describing the makeup of the individual pieces, and combining them together,
and I ended up with around six equations that take one step at a time to solve for th final exact width of the slat as well as I threw in a couple equations to solve for the exact length of a slat from short point to long point.
The quality of the video is poor focus at times, but I think there is enough information explained to show how this can be done, so if you have a project involving angular slats inside a frame, and are situated concentric with eachother, maybe this could be of some help.
Here is vedeo of a demonstration how I got these equations.
Here are the equations written here to better show the video details
TL= total length of frame
W= total width of frame
(a)= the chosen angle of the slats referenced from vertical (90deg)
N= the number of slats chosen to be in the frame.
eq1: EC=(TL / (2 x N))
eq2: O=(Tan(a) x (W / 2)
eq3: L=(EC – O)
eq4: ML=(2 x L)
these last two equations give the final width of the slats and the mitered length short pt. to long pt.
this value would be where to set the table saw fence to:
eq5: AW=(Sin(90 deg. – (a)) x ML)
read as (subtract your chosen angle from 90 deg. than take the Sine of that value, then multiply that new value by ML to aquaire the actual width of the slats.
Then finally and added bonus(haha)
the lenght from short pt. to long pt, of the slats is this
eq6: SL=(W / (cos(a)))
read as the slat length short point to long point is the Width of the frame divided by the Cosine value of yor chosen angle (a).
have fun in the shop…