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:
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 chord of the arc). Measure accurately the length of the straightedge and record that information, and also measure and mark the midpoint of the straightedge. We need to measure the deflection (line D in the diagram) that represents the maximum distance of the rail from the straightedge. Hold a ruler such that it is perpendicular to the straightedge and at the midpoint, then record that measurement (see photo). If your straightedge is short, you might have to use a Vernier caliper and measure in 1000ths of an inch. Be as accurate as you can, because slight variations in the measurements will introduce errors in the calculations.
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) + (S2 / (8 x D))
R = calculated result of the radius of the measured arc.
D = the maximum distance between the straightedge and the curve.
S = the length of the straightedge.
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.