When making trim for a circular or curved metal roof you must first measure the roof structure's radius of curvature. Usually contractors will attempt to trace a curve onto a piece of cardboard or scrap metal and then use that as a pattern. This works fairly well from a functionality perspective, but these curves are sometimes lumpy and irregular. A smooth constant radius looks better in sheet metal than an exact match random curve.
To measure the radius, you must collect two measurements from the curve you are measuring. One measurement is the chord length and the other is the perpendicular distance from the midpoint of the chord to the nearest point along the circumference of the circle. First, arbitrarily select a chord length that suits the scale of the job, perhaps 10% of the estimated radius. For anyone that doesn't remember this part of geometry class, a chord is a line bound inside a circle that connects any 2 points along the circumference of that circle. Once this chord is created, you will find the midpoint of this chord and measure the perpendicular distance to the nearest point along the circumference of the circle. So now let's let:
a = perpendicular distance from midpoint of chord to nearest point along circumference of the circle
b = half of the chord length
r = radius of the curve being measured
The formula for calculating the radius is:
r = (a^2 + b^2) / (2a)
My derivation of this formula and an example can be seen in the image below.
It is worth noting that it is difficult to build a quality radius. Because of this it is necessary to measure the radius in many places and ultimately select a radius that works best everywhere. If a radius changes too much from section to section, you may have no choice but to make patterns by tracing the random curvature onto pieces of cardboard or scrap metal. For constructing a radius, I have another mathematical solution for that too which will be shared in a future blog post.
I should also point out that although I typically use this formula for sheet metal work, it can be applied universally. So if you need to measure a radius, use this method.