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"Calculating Angle of a Triangular Facet?" Topic

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Thresher01 Supporting Member of TMP09 Dec 2020 2:10 a.m. PST

I'm interested in trying to make a Traveller-esque, Scout vessel, similar to their Type S model, though with more depth, since the original design is really too flat to have all that interior space frequently shown.

I'm thinking about using solid XPS foam for the project, instead of cutting bulkheads out, and gluing a skin on to that.

I need to calculate the angle of the faces of the outer portions of the model going with 4 x pieces of foam for the upper and lower, port and starboard pieces, and need to know what angle to cut them at.

The overall design will be 15" long x 10" wide x 4" deep (instead of the usual 3" depth at the rear of the hull).

Therefore, each piece will be 15" long x 5" wide x 2" deep (high), with 90 degree angles for the interior facets.

Of course, for myself and others, it might be helpful to be able to calculate the same design, with a 3" deep (high) design too, just for grins – each section being 1.5" in depth (height).

How do I calculate the cut angles needed in order to ensure that the outer, longitudinal lines match up properly?

I don't recall covering this in geometry class.

Here's a link to some images of similar designs, so you can get an idea what I'm going for, though again, mine will have more depth to it:


I especially like the sharp, pointed nose designs like this one:


zircher09 Dec 2020 7:37 a.m. PST

Interesting, what I did was inside out. I blocked out my deck plan in 3d and then wrapped the hull around that. It actually worked out quite well.

If you're interested in the papercraft…



Personal logo etotheipi Sponsoring Member of TMP09 Dec 2020 7:44 a.m. PST

You probably did it in geometry. It's just the basic Pythagorean theorem. The tricky bit would be in shop or mechanical drawing class (wherever you did orthographic projections), to figure out which triangles to calculate for each side. Each side of the dart face belongs to a different simple triangle on the "inside" of the ship.


The triangle you want, the surface of the "dart", shares one side with three simple right triangles that you know the other two sides for.


This "middle" side (the one across the top of the figure), is the hypotenuse of the length and half the height, best seen from the side view.


The "back" side is the hypotenuse of the half height, half width, as seen from the back view.


And the "outer" edge is the hypotenuse of the length and half the width, seen from the top view.



zircher09 Dec 2020 7:56 a.m. PST

I need to revisit my next iteration of that project where I split the hull into four layers so you could see the decks inside the ship. Got about 90% done and lost steam on it.

Three decks instead of two+split level. It's a chunkier design.


Stryderg09 Dec 2020 9:27 a.m. PST


etotheipi has a dastardly plan to take over the world by melting everyone's brain with mystical formulas and incantations…Pythagorean's Hypotenuses indeed!

Personal logo etotheipi Sponsoring Member of TMP09 Dec 2020 11:02 a.m. PST

You ought to see the math for calculating a hexagonal hexagonal-prism extrusion embedded within a toroid.

The new design works basically the same way. Break the complex shape down into bits you can handle. Then handle them.

If you're doing this on a 3D design tool, don't you have an "unfolder"? You should be able to find a script for your modeler that basically turns it into a card model and does the dimensions for you.

zircher09 Dec 2020 12:19 p.m. PST

One example of that is what I am using, Metaseqouia for the 3D model and Pepakura to unfold and cut it up for printing on cardstock.

Thresher01 Supporting Member of TMP09 Dec 2020 12:45 p.m. PST

I can calculate the length, as you say, but that doesn't really help me with the foam wire cutter.

What I really need to know is the precise angle of the green section (and the entire outer face of the model) in degrees, to set the cutter at, in order to make the first cut for the outer hull surface in the foam.

Once those cuts are done four times, I'll then cut the longitudinal one, and the rear bulkhead at 90 degrees to finish each piece.

I guess if there isn't a simple formula for that, I can always just draw a mockup, with 90 degree lines for the interior, and then measure the angle the old fashioned way, with a protractor.

Nice model, zircher! Thanks for sharing your images and files. I really appreciate it.

I considered using foamcore, and/or posterboard, or a mix of other materials, but I think XPS foam will probably be more durable, and perhaps easier to craft with too.

Personal logo etotheipi Sponsoring Member of TMP09 Dec 2020 1:23 p.m. PST


Sorry, this was what you wanted. Instead of calculating the length of the side, you calculate the angle as the arctangent of the two sides.

If you were cutting this on a tablesaw, I could recommend calculating the angle and dialing it in to the bevel.

Since this is a (tabletop?) hot wire foam cutter, just take the internal triangles, blow them up to 4 or 5 times size and use them to visually line up the appropriate angles.

Thresher01 Supporting Member of TMP09 Dec 2020 1:41 p.m. PST

Hmmm, that works for the rear end bulkhead, but the complicated part is getting a proper taper to a very fine point, for the bow of the vessel.

Making this cut, using a hotwire cutter, and turning the drawing 90 degrees on its side to do that, will create a nice, flat wedge shape, but doesn't permit me to get the gradual taper from the rear of the hull to the bow point, needed.

I was hoping to make a single cut for the proper shape, instead of having to make two cuts, one for each axis, in order to do that.

Now, my head hurts.

I suppose I should be able to do that ALL in one cut, if while sloping the cutting wire (and/or making a proper angle cutting tool – sloping the wire is probably easier), I angle the fence at the proper angle too.

Zephyr109 Dec 2020 2:45 p.m. PST

I'd cut out pieces of cardboard & tape them together to get a general idea of the shapes/angles before (making mistakes) cutting foam… ;-)

Ghostrunner09 Dec 2020 5:34 p.m. PST

Thresher have you tried using the law of sines?


If you know the lengths of the sides using the methods other people have suggested, it seems you should be able to use Law of Sines to calculate the interior angles of the triangle.

(side note: back when I was playing Traveler this one always got me too. A few years back I actually 3D modelled the Shout Ship to figure out the needed shape and the actual volume of the hull. As I recall, it was very squat and the tonnage was in the 130-150 ton range.)

Oberlindes Sol LIC Supporting Member of TMP09 Dec 2020 6:31 p.m. PST

A completely different approach would be to produce the ship in the software required for 3D printing, and have a squadron of them printed up.

Then you could go directly to painting.

@zircher: Nice work!

zircher09 Dec 2020 7:40 p.m. PST

Here is a handy tool that if you input the 3d coordinates of the three corners, it will spit out the numbers for ya.

I plugged in the numbers for a pyramid 15 tall, 10 wide, and 4 deep. Assuming 0,0,0 is the center of the back end.

Camcleod09 Dec 2020 7:58 p.m. PST

Don't know if this will quite make sense.
Take your 15" x 5" x 2" thick piece of foam, cut it into a triangle front to back 15" x 5" & 2" thick.
On the side edge draw a line from the front 0" to 2" high at the spine back end.
On the end draw a line from 2" high spine to 0" at the outer edge.
These two lines define the plane you want to cut on.
The back end at the spine will remain 2" and the outer edge of the triangle will be cut to 0"
Raise the outer edge of the triangle to 2" off the table so that the two lines you drew are horizontal and the back spine corner is sitting on the table.
Cut the foam horizontally along these two lines and the 0" outer edge.
Repeat for the other three quarters remembering that two of them will be the reverse slope cut.

Personal logo Saber6 Supporting Member of TMP Fezian09 Dec 2020 8:17 p.m. PST

Camcleod, this is what I had in my head but no words for

Personal logo etotheipi Sponsoring Member of TMP10 Dec 2020 5:46 a.m. PST

Hmmm, that works for the rear end bulkhead,

For one cut per surface, take the two jigs for the other two angles, and have them meet in a corner with the common length side. Then cut a triangle to make a "table top" on top of the corner. You will have a funky looking (scalene) pyramid. Then you put the foam on this, align the foam corner with the bottom of the hot wire (perpendicular to the table) and slide the long edge through, parallel to the edge of the cutter table.

You basically have to use the jigs to suspend the piece in space at just the right position.

Then reorient and cut again.

The foam block has to have the H and W of the final piece, otherwise you have to eyeball align the tip intersection. Very difficult to do. (You're using the corner of the block for alignment.)


I really recommend cutting the three jugs and making this with twelve cuts – three for each face. Larger amount of simpler work. This is especially good if you intend to do a fleet of them.

Personal logo javelin98 Supporting Member of TMP10 Dec 2020 9:15 a.m. PST

You ought to see the math for calculating a hexagonal hexagonal-prism extrusion embedded within a toroid.

No. No, I shouldn't. That way lies madness.

Thresher01 Supporting Member of TMP10 Dec 2020 1:12 p.m. PST

Thank you for ALL of the suggestions, hints, tips, and yes, even calculations done for me.

I really appreciate them.

Using the angled jig does sound like an excellent idea, however, given that the wire cutter is vertical, that presents another problem, since without gluing or somehow otherwise fastening the jig(s) to the cutting fence, it will be virtually impossible to hold them in place precisely.

If the wire was horizontal, it'd be a lot easier, though then you have to worry about wire sag was it heats up, I suspect.

I guess I could always try building a custom cutter with a wire horizontal to see if that will work, since presumably, if I can keep consistent pressure on the foam while pushing it through, then it won't sag or dip into the foam. There are instructions on Youtube to build them for low cost – worth checking out if you are interested in foam crafting.

Then, it would be possible to use jigs and gravity to aid me in construction.

I may need to go back to the drawing board on this, especially given the size I want, and might need to reconsider just using foamcore applied over internal bulkheads for the build. Right now, that seems like the easiest solution. I can then try to harden the foam, or apply a skin over it perhaps, possibly using thin styrene sheeting, etc..

Posterboard applied over foamcore bulkheads might also work too.

zircher10 Dec 2020 2:59 p.m. PST

Aye, if you go with thin styrene sheets over bulkheads, then the triangle calculator links that I posted should help with the hull plates.

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