How to Find the Radius of a Track Curve

Track curves may be thought of as an arc of a circle, so the radius is just the distance to its centre.

Measuring this distance for a curved section of track is problematic, because the circle is very large, so the centre may be on someone else's land, over fences, in a wood, down an embankment, on top of a cutting, in a river, etc.   Fortunately, Pythagoras gives a practical solution .....

In the diagram, the curve to be measured is at the top.   The radius we seek is r.   Use the running edge of the outer rail for all measurements.   We can measure the length of the chord c - this is the straight line between two points on the curve.   Accurately measure the small distance v between the centre of the chord and the arc of the circle.   This small distance is called the versine.

(Note that v is a small section of a radius - so the rest of the radius is r-v.)

Pythagoras applies to the right angled triangle, so:

r² = (c/2)² + (r-v)²

Expanding the bracketed terms

r² = c²/4 + r² - 2rv + v²

Cancelling r² & moving -2rv gives

2rv = c²/4 + v²

Dividing each term by 2v gives

r = c² / 8v + v/2

Because v is tiny compared with the length of the chord c, ignore v/2, so

r = c² / 8v

The radius of the curve is the square of the chord divided by 8 times the versine.