Calculating Angles

[Munroast]

I am making an Octagon pitched roof from wood.

At the base of the pitched roof the distance between the opposing pieces will
be 290mm (external)

The height (vertical) will be 500 mm

What angle do I need to bevel the sides of the bits so as they fit snugly together, and how would you calculate it.

Spoiler

this is what I got 18.5 degrees, but really not sure how to work it out

Just to be clear what angle I'm after

 

[big-all]

usually its trial and error but often your saw wont be up to the compound angle
or there wont be a big enough bit left to secure safely
what saw are you trying to use?

 

[Munroast]

I can push it through a planner that has settable angles and finish it off with a belt sander that has adjustable guides.

I agree there is often a degree of trial and error with these things, but it is best to get it as close as possible on the first go.

But more than that I am curious as to how to calculate the angle - I have come up with a figure in the OP but not sure if I have used the correct method. So in many ways it is a DIY maths question rather than a DIY woodworking question

 

[Mr Chibs]

Don't know if this helps?

 

[Notch7]

It would depend on the roof pitch.

 

[Munroast]

It would depend on the roof pitch.

indeed, that is why I gave the height and width.

 

[dontbelieveawordofit]

I can push it through a planner that has settable angles and finish it off with a belt sander that has adjustable guides.

I agree there is often a degree of trial and error with these things, but it is best to get it as close as possible on the first go.

But more than that I am curious as to how to calculate the angle - I have come up with a figure in the OP but not sure if I have used the correct method. So in many ways it is a DIY maths question rather than a DIY woodworking question

you’ll just have to wait until johnd or bobbydazzler research it on the mathematicalscholastic forum and come back with the answer as if they’ve been doing it all their lives.

 

[StephenStephen]

You could draw it out to scale and measure it as a second check

 

[foxhole]

indeed, that is why I gave the height and width.

You have a two dimensional roof?

 

[Munroast]

Don't know if this helps?

thanks for this - pity he didn't say how he did the calculations, but never-the-less he picked up on the same principle that I used
see the bit with the the piece of paper 0:56 to 1:20
at verticle (90º) the angle would be 22.5°
when flat (0º) the angle would be 0º

so therefore if the pitch was 70º then the angle would be 70 / 90 x 22.5 = 17.5º
so as per mine the pitch would be 73.8º so therefor 73.8 / 90 x 22.5 = 18.45º

till don't know if that is the correct way of doing it?

 

[Careful_Bodger]

Ooooo - this is an interesting one.

Firstly, I don't think you can scale between 0deg and 22.5deg that way you have. It looks like you have assumed a linear variation and you have done (not unreasonably) what's called linear interpolation.

If you look at the mathematical formulas in the spreadsheet (https://woodgears.ca/miter/splayed_miters.xls) this does show how he did the calculations. There are complex formulas of tangents (TAN), sine (SIN) and cosine (COS) of the angles. While I cannot see at the moment how he got those, please read on.

If I understand correctly, roof pitch = 90deg - splay angle, where splay angle is shown on his website (https://woodgears.ca/miter/). Tilt & mitre angle are also shown in the same diagram.

So your (fairly steep) 73.8deg pitch = 16.2deg splay. Is that right?

Looking this up in his downloadable PDF (https://woodgears.ca/miter/splayed_miters.pdf), the nearest result is 16deg in the left column, and reading acoss to the bit with 8 sides & flat mitre angle 22.5deg, then I reckon that's actually 21.58deg for the saw tilt angle and 6.51deg for the mitre.

The difference for 16.2deg splay compared to 16deg splay is very small, and will be negligable, I am guessing, compared to cutting tolerances.

You will also have the lower edges which will also need a (mitre...oops, nope!!) tilt cut at the splay angle.

I am quite interested this (mathematically), so please comment if I have got the numbers wrong & you need a recalculation.

 

[mrrusty]

This is related to the settings you need to cut a coving "on the flat". I have an excel I use - here are the formulas - don't ask me to explain them, I copied this from somewhere.

spring angle in this calculator is the angle between the back of the coving and horizontal when the coving is installed (some calculators show the spring angle as the angle between vertical)

Have a look at your mitre saw - it probably has presets marked at 33.9 and 31.6 degrees - I bet many people don't know why! - its the tilt and mitre for cutting "standard" 38/52 degree coving "on the flat" for a 90 degree corner

 

[Careful_Bodger]

Better than related Sir Rusty!

These equations provide the same answers. With the OPs approximate pitch = spring = 74deg and 'corner' = 135deg, I get the same answers as I mentioned previously.

Someone very clever with 'trig' has somehow simplified these, it seems.

 

[paulrockliffe]

I could work the angles out but I'd never trust myself not to make a mistake. I would draw it in SketchUp and measure the angles there.

 

[Munroast]

Ooooo - this is an interesting one.

Firstly, I don't think you can scale between 0deg and 22.5deg that way you have. It looks like you have assumed a linear variation and you have done (not unreasonably) what's called linear interpolation.

.

Yes, I had a very big suspicion that it would not be linear (hence me asking) all these things revolve around sine cosine and tangent and they always create curved lines on graphs and my theory would create a straight line. So I sort of guessed I was wrong.

I think @mrrusty excel has the answers


It is certainly a very complex bit of maths, I must try and get my head round sometime, too complicated the night though - esp whilst on the home brew.

In the mean time I can just use Rusty's excel thing.