Calculating angles . . .

Okay, guys . . . . . give me a hand with a trig problem:
I'm going to build a small stand. The base will be a tapered cone. Dimensions are 18 inches high; 8 inches diameter at the top and 12 inches diameter at the base. I'll glue up 6 tapered boards into a hexagonal cross section, chuck it into a lathe and turn it smooth.
If this were a uniform column, the boards would join at an angle of 60 degrees. The taper adds an embuggerance. Any suggestions as to how I would calculate the angles?
I've done this before, but can't remember if I used a trig solution, a graphic solution or a chart from Lahee's "Field Geology" (true dip vs apparent dip).
(My copy of Lahee got eaten up by a garage sale along with my g-pick and brunton before a move a long time ago.)
Suggestions?
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Take a look here for a trial version
http://www.perfectcuts.com/index.asp
Also good for flower pots and planters.
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Try this also if you want a freebee.
http://www.woodworkersguildofga.org/ShopHelpers/MiterCalculator.htm
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and this shows the formulae, if you're interested.
JeffB
Leon wrote:

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Sheesh, kind of a tall order :-) The math is a little long to explain here, but you can see it if you think of the hexagon as a series of six equilateral triangles, each one of which is a 30-60-90 triangle having sides 1 and sqrt(3) and hypotenuse of 2. Anyway, here's what I got:
- Individual pieces are still mitered at 60 degrees. You can see that by considering the circle cross section anywhere along the length of the cone.
- The six individual sides have a minimum length of each of the bottom edges of just under 7" (6.928" = 6/sqrt(3)*2). Assuming excess to turn off, make the bottom edges of each side 7-3/8" on the wide edge of the miter. Then make the length of each side 18.11" (just a tad under 18-1/8"). The top length should be 4.91". The taper jig angle should be about 86.1 degrees. And the stock has to be pretty thick, at least 1.7" thick if you want a minimum 1/2" wall thickness.
Anyway, I can manage the math, but I don't know diddle about the turning. I would sure do a cheapie mockup before committing any good wood ;-P
Cheers, Nate
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nit #1 -- 'equilateral' and '30-60-90' are mutually exclusive.

I'll buy: 'equilateral triangles, all sides of length 2'. each of which can be sub-divided into a mirror-image pair of 30-60-90 right triangles, of the dimensions you give. <grin>

Nit #2: The angle of the join, measured _horizontally_ is 60 degrees, agreed. The angle, relative to the plane of the board, is different.
This is a similar issue to compound miter cutting for fitting molding into corners.

Something looks 'off' in that calculation: a 12" (across the diagonal) hexagonal base, has a 6" radius dimension (from center to an outside corner). from center to the (outside) middle of a side will be (sqrt(3)/2)*6", which evaluates to 5.196". Thus, the source stock has to have that 0.804" plus whatever finished wall thickness is needed. for 1/2" wall, it'd be 1.304"

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Robert Bonomi wrote:

dave
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For correctness [and more nitpicking...] Properties of scalene triangles are properties of all triangles. The definition is the generic triangle with nothing in particular to otherwise identify it; all sides unequal, no right angle. The six triangular sections of a hexagon are equilateral though, having all sides equal. Four sides would give right triangles; in fact, right isosceles triangles. Five, or more than six, will give isosceles triangles. Six is also isosceles, but moreso; being equilateral it is isosceles three ways.
Bill. :-)
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A 'Microsoft tech support' response -- "technically accurate, but useless".
The original poster did _not_ use the ter 'scalene', so it isn't fair to use it in pointing out the inconsistencies between the particular terms the OP _did_ use.
"Thank you for playing."
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On Wed, 16 Jun 2004 02:32:30 +0000, snipped-for-privacy@host122.r-bonomi.com (Robert Bonomi) wrote:

etc....
It's a wonder they haven't invented the computerised saw. Just punch in the number of sides, angle of sides etc., and the blade and miter adjust themselves. Perhaps they have, and it's not in the Orange Giant yet?
Bill.
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There *IS* a whole _class_ of that kind of equipment.
It's called 'CNC tooling'.
The machine itself just takes a list of positioning and cutting commands.
"Front-end" systems, however, can take a 3-d representation of the part to be made, and the rough stock, and 'figure out' the entire command-list to "remove everything that's not part of the thing being made".
The things those kind of machines can do _is_ downright scary.
As for this particular 'problem', any half-way decent '3-D' CAD program will let you start with a vertical plank, tilt it to the desired angle, chop the top and bottom off parallel to the ground, and chop the sides to a vertical radial from the 'center'. Viola!, you've got the object you need. add some 'centerlines' for reference, and you can just have the software 'read' the angles needed.
The same approach works for compound miter settings for molding corners - 'the corner looks like _this_', the boards go like 'this', 'the join is the plane _here_', and 'the _back_side_ of the molding is like _this_.'
Then pull the piece of molding out of the model, rotate it so it's flat on it's back, and just have the s/w 'show' the angles. set the saw to match, and "awaaaaay we go!".
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snipped-for-privacy@houston.rr.nospam.com says...

Take a trip over to the Lee Valley website, and navigate to their router bits for birdsmouth joinery. . .Once you get there, there is an online copy of the use instructions that has lots of useful stuff on making tapered cones. Near the end is a table that has degree of taper and everthing!
Kim
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