Hello Mike,
To answer your question about the math… This is what I used to teach in the Coast Guard Auxiliary to Coasties and the public.
When anchoring, we essentially have a right triangle with the depth as a side of our triangle, the anchor rode as the hypotenuse, and the swing radius as another side.
Reviving some long dormant brain cells from my university days…
We can solve this using the Pythagorean Theorem which states that if we know the lengths of any two sides of a right triangle, we can find the length of the third side.
The formula is (and apologies for the poor mathematical symbology as I can’t figure out how to get this correct in this interface.):
Hypotenuse = sqrt(side^2 + side^2).
Side = sqrt(hypotenuse^2 - side^2)
For example… If we are anchoring in 40’ of water and the bow roller is 10’ above waterline… Then the depth is effectively 50’. So, we put out 150’ of rode to get 3:1. We want to know our swing radius.
We use the formula, Side = sqrt(hypotenuse^2 - side^2).
40 (depth) + 10 (height of bow roller) = 50’ for one leg of our triangle. The hypotenuse is 150’ (our rode).
Side = sqrt (150^2 – 50^2) = sqrt ((150*150) – (50*50)) = sqrt (22,500 - 2,500) = sqrt (20,000) = 141.42’
So, our swing radius is 141.42’ Not significantly different than the 150’ rode we deployed.
If we take a deeper example, say 90’ depth, 10’ bow roller height and 3:1 scope then:
Side = sqrt (300^2 – 100^2) = sqrt ((300*300) – (100*100)) = sqrt (90,000 – 10,000) = sqrt (80,000) = 282.84’
Again, the swing radius isn’t significantly different from the rode we deployed.
All the math notwithstanding, I use the rode we deploy as the swing radius when I set my anchor alarm.
I set the alarm from the pilothouse when the anchor touches bottom and adjust the alarm circle to match the amount of chain I put out plus some margin to reduce false alarms (maybe another 20%).
I hope that helps in answering the core question you asked.
The big takeaway is that at 3:1 , 4:1, or even 5:1 (for your east coasters), the depth doesn’t really matter when figuring swing radius. Just use the amount of chain you put out as an approximation.
Safe Adventuring,
Peter C. Adams
m/v Blue Redd 5726