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
(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.)
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 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
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
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"
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.
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."
On Wed, 16 Jun 2004 02:32:30 +0000, firstname.lastname@example.org
(Robert Bonomi) wrote:
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
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!".
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
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