Laying out a helix as a triangle is not all that difficult, but it’s a little bit more involved than an oval. I laid mine out using simple geometry tools.

I started out by figuring what radius I wanted to use, and how long I wanted my straight sections to be. Once I had an idea about that I made a cardboard model to help me figure out the angles I needed. For my layout I chose a 15 inch radius, two 25 inch long straight sections, and one 6 inch long straight section.

On some box cardboard I grabbed out of the trash, I drew three ovals at a scale of 1mm to 1 inch. The ovals need to be pretty small so you can find the angle on a regular sheet of paper. I set the compass at 15mm, and using a ruler I drew two circles 15 mm apart with the centers along a line. Did the same to make another oval. Then I made a smaller oval by drawing two circles with their centers 6 mm apart. Using a ruler I drew lines across the tangents of the circles. That done I cut them out closely with scissors and ended up with these:

Next, I pinned all three of them together through the centers with pushpins:

I taped the three pieces together to create a template from which I can determine the angle to cut the roadbed of my helix to make a triangle out of a circle. First I drew a straight line across a sheet of paper. I laid the template on this line so that one of the long straight sections lined up with the pencil line.

Holding the template in place, I laid a ruler along the other long straight section and drew a line against the ruler where the ruler intersected the original pencil line.

Now I can measure the angle of the intersecting lines with a protractor.

Now that I have this angle determined, here is how I laid out my roadbed to make the triangular helix:

The circle is cut up into segments. The segments get rearranged, and the straight sections are added in to come up with the triangular oval shown here:

Two of the straight sections of my helix were the same length. That meant I only had to find one angle to cut the circle to make it a triangle. If all three straight sections are different lengths, you’ll need to find two angles. It doesn’t really matter which two, just pick two and the third angle will take care of itself.

One other point to mention, and one I didn’t realize until after I committed to building this beast, is that a triangular helix doesn’t really save much space when you compare it to a round one of a larger diameter:

Still, it saves some, and if you want a turnout in your helix, having a straight section is the easiest way to incorporate one. Unless you handlay your turnouts, of course. Perhaps a triangle of a different shape might fit into an odd spot easier than an oval or a round helix.