Gaylord Gill prepared this article and its accompanying diagram and photo. He graceously shared it with us for us to enjoy. Text and images are copyright © Gaylord Gill.

Most modelers understand that certain equipment (steam locomotives in particular) won't operate around a curve that's too tight. So for the equipment you plan to run, it's good to know the minimum radius needed on your trackwork.

There are a couple situations that might arise in which you'll want to calculate the radius of an already-existing curve:

- You laid your own curves freehand without measuring the radius, and now you're thinking of buying a locomotive that requires a minimum radius of, say, 36".
- You have a manufacturer's turnout of a particular frog angle, say a #5, and before you install it, you want to know if your equipment will go through it.

Refer to the diagram, which shows the arc of a curve of unknown radius (line * R*). Select a straightedge of convenient length, the longer the
better. It needs to fit fully within the curve you're measuring, with the assumption the radius is the same throughout the arc. For example, if you're
determining the radius of the divergent leg of a turnout, your straightedge might have to be six inches or less.

You're going to use the straightedge to touch the arc of the track at two points (line * S*; in geometry, this is called the

You've now taken two measurements, and if they are in fractions of an inch, you'll find it helpful to convert them to their decimal equivalents. Using a calculator or computer, plug your measurements into this formula:

R
=
(D / 2)
+
(S^{2} / (8 x D))

where:

* R* = calculated result of the radius of the measured arc.

Let's run through an example. Suppose we have a length of rigid stripwood that's exactly 24" long, and we have placed a mark on it at its midpoint. After arranging our straightedge against the inner rail of our curve, and laying a ruler against it as described above, we measure the deflection of the rail at 1-45/64". We then convert the deflection measurement to its decimal equivalent: 1-45/64" = 1.703" (two-to-three decimal places is precise enough).

For the first component in our formula, we calculate "D / 2", or 1.703 divided by 2 = 0.852.

For the second component, S-squared will be 24 times 24 = 576.

For the denominator, 8 times D = 8 times 1.703 = 13.625.

Then 576 divided by 13.625 = 42.275.

Finally, we add the two components: 0.852 plus 42.275 = 43.127.

So the calculated radius of our curve in this example is just over 43", a fairly broad
curve for S scale equipment.