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Fair winds!

Dave and Anke
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Saturday, February 20, 2016

Magic Triangles: Handy Math Trick for Befogged Brains

Mathematics is a collection of cheap tricks and dirty jokes.
-- Lipman Bers 

Magic Triangles: Handy Math Trick for Befogged Brains

Magic triangles are a handy little mental trick that helps with some of the math we boaters frequently encounter.

Anything that makes math simpler helps in those wee, hypothermic hours of pre-dawn fatigue! Anything that makes math simpler helps cut down on error, even when wide awake!

I first read of these in Miner Brotherton's 12V Bible for Boats (one of the most accessible DC12V sources around), who learned them in the Army. Apparently, some go so far as to tattoo them on the back of one hand.

Magic Triangles work for any equation of the form X = Y x Z.

Consider the Speed, Distance, Time calculations a navigator must face, if only in the rough. There are three, related formulas:

D = S x T ......... If we know Speed and Time, we can find Distance.
S = D / T .......... If we know Distance and Time, we can find Speed.
T = D / S .......... If we know Distance and Speed, we can find Time.

Got that? I don't! This kind of thing doesn't stick in my brain.

Each of these is a variant of the others, transformed by simple, algebraic balancing acts. Right.

I used to remember that car Speed is given in familiar miles per hour (aka Distance / Time), so S = D/T, and then work the others out as needed. But this doesn't come easy in the bleary hours, and paper helps at the best of times.

Here's the equivalent magic Triangle:

To use, cover the one you WANT TO KNOW. Of what's left showing, one over another means divide; side by side means multiply. 

Let's say you want to know S. Cover it with a finger, and we see D/T. Shazaam!

Three formulas in one. Better yet, I find I can do this in my head without breaking a sweat.

Any mnemonic works that puts the top letter first, followed by the other two (their order isn't important... DST or DTS works the same). 

What's important is that you start with the variable that stands alone, and equal to two multiplied variables. 

EXAMPLE:   Xamine Your Zipper    =>    X = Y x Z    =>    Magic Triangle with X on top.

Here's a couple others that are handy afloat or ashore:

These handle most marine electrical calculations.

So there ya go... MAGIC Triangles... making smart fellers of fart smellers since 1906.


Extra Credit:

As with a spreadsheet, you're not limited to a simple variable within triangle spaces. Consider the following, common navigational trio:


Angles can be measured by sextant, compass, pelorus, cross-sticks or fingers. Distance may be known from fixes on the chart. Spread may be known from the chart, topo maps or average heights (an example in our area is trees, for whom average height is about 100ft). Other clues may be pressed into service, such as roughly 10 vertical feet per story in modern house construction. Tangents can be looked up in tables, on the web, using scientific calculators (many phones can handle these).

All subjects covered elsewhere (though not by me). 

Go get 'em, Tigers!

PS. Astute reader JOHN caught me with the wrong magic triangle for the extra-credit (now repaired)... thanks, John!

As it turned out, I'd already realized it, as the same afternoon, Anke and I used it to lay out our sails, but it wasn't making any sense!

Just goes to show, easier is still a challenge for some of us!  8)  


  1. Very nice! Now what to do with it??! Errrr...where are the headache pills? Hope all is well in your neck of the woods Dave. Boat building has ground to an ignominious halt here. Hospital stays and visits, and chemotherapy has seen to it. Hopefully things will return to some sort of normal sometimes soon. Joel

    1. Hi Joel,

      I'm so sorry to hear of your health issues! We'll be lighting the joss sticks for you.

      Fair winds and full recovery!

      Dave Z

  2. Joel: "Very nice! Now what to do with it??! Errrr...where are the headache pills?"

    My thought exactly. I was chatting with some solar nerd people who were trying to tell me how simple and straight-forward it is to calculate whatever. To prove their point, they started by saying, "OK, just take _W_atts, which we'll call 'P', and _A_mps, which we'll call 'I' and then ..." The most infuriating part is that they seem oblivious to why that is a problem at all. Makes me want to bonk them on the head with the closest heavy object.

    In the case of the triangle with "tanA", I assume that the most likely practical use is to estimate more accurately a visual horizontal distance, for example, across the water?

    Dave, thanks for trying to make a bad thing for some of us a bit better.

    1. Hi Yoda,

      Yeah, symbol proliferation is a mathematical irritant.

      Current being 'I' drives me especially bonkers. Stands for the French phrase "intensite du courant". Don't have to use it, but you'd end up out of step with everyone else. Just hafta bite the bullet.

      If you prefer, WAV and VAO (Watts, Amps, Volts and Ohms) can be used in the MTs, working directly in units.

      The DtanAS MT helps figure height OR separation and distance off. Horizontal OR vertical. Depends on what you know, going into the calculation.

      Dave Z