Check my work - 8 sided pyramid - finding miter cutting angle

[Brian Kent]

As a part of a large candle-holder project I am making an 8-sided pyramid. I need to figure out the table saw blade angle for the miter-cuts. If you know how to do this, could you check my work?

I am using this calculator:
http://www.pdxtex.com/canoe/compound.htm

I started with the slope calculation. The pyramid is 9" across at the base, so the Horizontal Run is 4.5". The Vertical Rise is 18.75". The resulting slope is 76.5° (76.504 to be exact).

It has 8 sides so the included angle shows as 135°. That calculates to an end angle of 5.5° (5.522) and a bevel angle of 22° (21.84).

Does this seem reasonable, before I try it on lumber?

[Andrew Hughes]

If you can figure out how to cut those parts on your tablesaw I would be impressed more than the math that got you there.:confused::) Looks very tricky.

[Jamie Buxton]

I tried a completely different method. I made a Sketchup model, and measured the angles.
I'm guessing your 9" and 4.5" numbers are to the middle of the flats of the octagon. That is, from tip to tip of the octagon would be a little larger.

If I look at one of those facets, I get 11 degrees at the tip, not 5.5 degrees. A factor of exactly 2 seems suspicious. You can probably do a rough check on this angle from your paper model. Other than that, my method agrees with you -- 21.9 degree bevel, but you can't set a table saw bevel down to the tenth of a degree.

[Myk Rian]

I make my 8 sided light houses using a 22.5º router bit on the edges.

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[Dave Richards]

Similar to what Jamie came up with, also in SketchUp

https://farm1.staticflickr.com/624/22504525636_8e4c8c0f02_z.jpg (https://flic.kr/p/AhDzqG)
Laying it out graphically helps to make sure the numbers are right.

[Chris Padilla]

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Here is what I got in Sketch-up. 22.5 degrees for the octagon.

84.9 degrees for the isosceles triangle's large angle and 10.2 for sharp, acute angle.

I assume 1/2" stock in the diagram, FWIW.

[ken masoumi]

I make my 8 sided light houses using a 22.5º router bit on the edges.

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And that jig you posted in your thread would be perfect for this project ,all it needs is a little tweaking here and there.

[Jamie Buxton]

Here is what I got in Sketch-up. 22.5 degrees for the octagon.

84.9 degrees for the isosceles triangle's large angle and 10.2 for sharp, acute angle.

I assume 1/2" stock in the diagram, FWIW.

So you went 9" point-to-point on the base octagon. That might explain the differences from my version, which is 9" flat-to-flat on the base octagon.

[Ole Anderson]

21.9 degree bevel, but you can't set a table saw bevel down to the tenth of a degree.

Just use one of those Wixley magnetic digital gages set on the blade, zeroed on the saw top: http://www.japanwoodworker.com/Images/products/400/159488.jpg

[Gene Davis]

This one is a lot less. Maybe as good? I don't know.

[Dave Richards]

Or the Angle Cube (http://www.eagleamerica.com/product/424-0128/shop_by_brand_-_igaging) from Eagle America is on sale for $24.99.

[Chris Padilla]

So I'm curious to hear whose measurement/calculations were the closest, Brian! :D

Let us know when you build it.

[Pat Barry]

It seems to me that if you were making an 8 sided barrel and not a cone, then its an easy answer that 360 / 8 / 2 = 22.5 deg miter angle for each side of each stave if you will. However, when the barrel is turned into a cone, that 22.5 deg angle changes, right? The thicker each board, the more of an affect it will have. You would need to look at a cross-section through the board that is perpendicular to the face of the board. Too much math for me right now

[Dave Richards]

...that 22.5 deg angle changes, right?... Too much math for me right now

You're right. The angle is not 22.5° Too much math for me, too. That's why I made the drawing I posted. :D

[Chris Padilla]

That might be correct but I don't understand why. If the footprint of this cone/pyramid/whatever-it-is is an octagon, then it is an octagon no matter where you slice it and no mater how far the sides tip in to go from a cylinder/barrel to a cone/pyramid. So that should be 22.5 degrees.

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The way I made this is Sketch-Up was to start with an octagon, expand up the sides to create a cylinder/barrel, and then tip the sides in. I just don't see where that would change it from 22.5 degrees.

[David Winer]

As a part of a large candle-holder project I am making an 8-sided pyramid. I need to figure out the table saw blade angle for the miter-cuts. If you know how to do this, could you check my work?

I am using this calculator:
http://www.pdxtex.com/canoe/compound.htm

I started with the slope calculation. The pyramid is 9" across at the base, so the Horizontal Run is 4.5". The Vertical Rise is 18.75". The resulting slope is 76.5° (76.504 to be exact).

It has 8 sides so the included angle shows as 135°. That calculates to an end angle of 5.5° (5.522) and a bevel angle of 22° (21.84).

Does this seem reasonable, before I try it on lumber?

This is a fairly common problem. I have made octagonal bird house tops and slanting-side flower boxes using a BASIC program (I actually wrote!) as an "app" to do the math. That was a long time ago and I no longer have the program or the skill to write it. However, I have collected programs (better than mine) that do this math for woodworkers. An easy one to follow is from the Woodworkers Guild of Georgia, see http://www.woodworkersguildofga.org/ShopHelpers/MiterCalculator.htm Using this program with your 76.5 slope angle and 8 sides I get blade angle 21.8 and miter angle 5.72 (90-84.28).

[Dave Richards]

That might be correct but I don't understand why. If the footprint of this cone/pyramid/whatever-it-is is an octagon, then it is an octagon no matter where you slice it and no mater how far the sides tip in to go from a cylinder/barrel to a cone/pyramid. So that should be 22.5 degrees.

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The way I made this is Sketch-Up was to start with an octagon, expand up the sides to create a cylinder/barrel, and then tip the sides in. I just don't see where that would change it from 22.5 degrees.

Chris, you need to measure the bevel angle perpendicular to the edge getting beveled. I'll make a drawing to show that.

Edit to add:
https://farm6.staticflickr.com/5768/21932823693_04c13b96c5_z.jpg (https://flic.kr/p/zq8smD)

The "blade" is leaning on the bevel. You have to measure that angle perpendicular to the path line, not the one at the bottom.

Consider an extreme case in which the octagon is laying flat. The bevel angle will obviously be 0°. As the height of the structure increases, the bevel angle will increase toward 22.5° but it won't actually reach that point until the sides are vertical.

[Bill Orbine]

That might be correct but I don't understand why. If the footprint of this cone/pyramid/whatever-it-is is an octagon, then it is an octagon no matter where you slice it and no mater how far the sides tip in to go from a cylinder/barrel to a cone/pyramid. So that should be 22.5 degrees



It's compound angles. Think of it like cutting a 45 degree miter on a crown molding laying flat on the compound miter saw. Typically, a 38/52 crown will have 33.9 degree bevel and 31.6 degree miter setting on the compound saw to achieve a 45 degree miter on the crown. It's the same concept, but a bit more complicated, when cutting these 8 pieces for the octagonal cone FLAT on the table saw, in this case 21.9 degrees. However, the foot print of the cone still looks 22.5 degrees when the cone is assembled just like as you look straight up towards the ceiling viewing the 45 degree miter on the crown molding.

[Brian Kent]

Thursday is my weekend, so I will be trying it out. I will be making a sled and adjusting the blade angle until it is perfect (enough).

[Eric Schmid]

If you can figure out how to cut those parts on your tablesaw I would be impressed more than the math that got you there.:confused::) Looks very tricky.

Exactly! I recently pickup up a new miter saw and cut these as part of the set-up and alignment. I was able to get the fence to within 2 thousandths of an inch from square to the blade. If my math is correct, and this is not my strong point, the saw is out .016 degrees or so. After four sides were put together, the closest I could get to closed on all miters was 1/64. Add in the compound cut and the potential for compound error and it does sound like a finicky project. A custom sled and a good miter gauge would get pretty close. I'd probably try it on a track saw or do what Myk suggested and use a router bit after cutting the tapers.


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[Brian Kent]

The preliminary cut is done. Next I will test fit and trim as needed. Much more time went into math and thinking about sequence of cuts than actual cutting.

[Brian Kent]

And the answer is 21.8°. I glued up the first of 2 pyramids yesterday.

The margin of error gap was .038" (1/26") out of a diameter of 28" at the base. That works out to about 1/2°. Divide that by 16 cuts and you have about 1/32° error per cut. I had set the saw to as close to 21.8° as possible and got a little bit lucky too.

I will deal with the .038" gap with a small wood-wedge.

[Jim Seyfried]

I will deal with the .038" gap with a small wood-wedge.

Could you glue up two groups of four, true up the halves and then glue the halves together?

[keith micinski]

Could you glue up two groups of four, true up the halves and then glue the halves together?


I agree with this. I would make two sides and then take a block plane and true the 4 faces up that way making the angle a littler greater then it needs to be since a gap on the inside of the project can't be seen like the gap on the outside of the project. I probably would have set my saw up this way also

[Andrew Hughes]

Great job Kent,Very meticulous with your cutting and setup.Nice even gap.
I wish I had tablesaw skills like that.

[Brian Kent]

Thank you very much, Andrew.
Jim, I could do what you said, and still may with the 2nd pyramid. My hesitancy is that I would need to glue many sheets of sandpaper to true a piece that is 9" x 16".
Keith, I would follow your suggestion if this were the final step. Net comes the lathe work. Half-way up the side the curve will go in almost an inch, so the hollow-joint trick wouldn't work.

[keith micinski]

Brian can't you just take a block plane and true up the faces that meet with few passes and a few test fits?

[Eric Schmid]

Nice work Brian! How did you set up your taper sled at the desired angle?

[Brian Kent]

Eric, I started with the angle for the bottom of the triangle, screwed it in, and then did the long side. All of the angles are from the math and modeling done above. The side block is the same depth as the cut piece, so I could double check by sight and tough to be sure it was level.

Another important measuring step - with the blade set at 21.8°, I measured the distance from the edge of the sled to the blade, at the height of the piece I was cutting. By luck it was exactly 1", so I could measure the side block on the sled with that 1" accounted for.

I am very happy with the second glue-up. No visible gaps. I had left one wedge un-glued so that I could adjust the angle if needed, but it looks like I can just glue it in when the rest of the piece sets.

[Chris Padilla]

Nicely done, Brian! I love getting all detailed, anal, and fussy with such things. But I sure do "waste" a lot of time doing it. I still need to wrap my head around that bevel angle NOT being 22.5 degrees but one day, I'll get it. :)

[Kelly Craig]

I'm having trouble seeing the problem. When I was starting out and cut my first triangle, forty years ago, I started to sweat the angle capability of my saw, but I got over that after I realized I could stand the work up to increase the angle. Obviously, getting 120 degree angles with 60 miters was no problem, when approached from 30 degrees.

To do something like this, I would consider a sled and a clamp system. I'd probably make it adjustable, like I did my band saw diamond cutter, so it could be used on any unique project calling for compound cuts.324935