Some time ago, the discussion on π being the incorrect number for calculations in trigonometry, in favor of τ (2π), led me to muse about creating a unit to replace the degree, called the wedge and being equal to nearly two degrees (100^{W} = 60°).

I found myself musing about the topic again, but I’ve decided that the wedge actually makes the problem worse. One issue with degrees is that the numbers seem fairly arbitrary, but another is that the numbers are too big.

Somehow, I have reached this elder age without realizing that the sum of the interior angles of a triangle is π radians. Like the degree, the radian is a measure of rotation. But while students struggle with understanding why there are 360 degrees in a circle, they struggle even more with the relationship between π and radians.

### Radians

If you have circle with radius r, a radian is the size of the angle created by moving r along the circle. For instance, if you have a circle that is 20 feet in diameter, a radian is the angle formed by traveling 10 feet along the edge of the circle.

When it’s described that way, it’s not terribly confusing. But seen as an amount of turn, without reference to the ratio between radius and circumference, a radian represents a turn of \(\frac{1}{2\pi}\). Because this doesn’t make much sense without the context of a circle, students tend not to get exposed to radians when first studying triangles.

Because π is so connected to radians (the length of a circumference is 2πr, so there are 2π radians in a full turn of a circle), it is often convenient to treat π rad as if it’s a unit unto its own, even though π is properly speaking a number.

### Piradians

There is no reason we can’t simply create our own units. All of these units are arbitrary anyway. What’s an inch? What’s a pound? Units were originally created in order to measure things consistently, and it’s important that “a meter” always refer to the same distance, but what that distance is is fairly arbitrary (One meter is the distance traveled by a ray of electromagnetic (EM) energy through a vacuum in 1/299,792,458 (3.33564095 x 10^{-9}) of a second.).

If we find ourselves consistently referring to fractions of π with the unit of radians, we can create a unit of a piradian. A piradian is half a turn around a circle. The sum of a triangle’s interior angles is one piradian. The sum of a square’s interior angles is two piradians.

It’s easy enough to see why there was no transition from radians to piradians. Anything to do with triangle angles would be in fractions. There is one very nice part about using piradians: If \(0 \lt \theta \lt 1\), then:

- \(\sin{\theta} = \sin{(1-\theta)}\)
- \(\cos{\theta}=-\cos{(1-\theta)}\)
- \(\tan{\theta}=-\tan{(1-\theta)}\)

This is a little clearer and hence easier to remember than the standard radian rules:

- \(\sin{\theta} = \sin{(\pi-\theta)}\)
- \(\cos{\theta}=-\cos{(\pi-\theta)}\)
- \(\tan{\theta}=-\tan{(\pi-\theta)}\)

In short, using the piradian as an angle unit would prepare students for using radians before having to deal with π. To convert from piradians to radians, simply multiply by π.

However, the piradian suffers from the same problem as the gradian: Two common angles are awkward in base ten. 30° = 1/3 piradians, while 60° = 2/3 piradians. Granted, 45° = 1/4 piradians and 90° = 1/2 piradians. Still, though, all those fractions can be troublesome.

### The Spry

In order to cover the four common triangular angles, the greatest common factor is 15. This represents one-twelfth of a piradian, suggesting a mouthful of a name, semisextanspiradian, and a jaunty abbreviation, “spry”.

Let us henceforth pretend no knowledge of the mouthful and embrace the abbreviation. The plural shall also be spry.

So one spry is equal to 15° and π/12 rad. Here are some standard conversions:

Degrees | Radians | Spry |

30 | π/6 | 2 |

45 | π/4 | 3 |

60 | π/3 | 4 |

90 | π/2 | 6 |

180 | π | 12 |

360 | 2π | 24 |

The oft-taught special triangles are 2-4-6 and 3-3-6. Right angles are 6 spry. A line is 12 spry.

This unit allows for much more reasonable numbers for students while allowing for an easy transition to both angles (multiply by fifteen) and radians (multiply by π/12).

Clocks could also be used to teach spry sense: Each hour represents two spry because we go through the clock of twelve hours twice a day.

Of course, in practice, there is need for finer granularity than 15° angles. A decispry (dspry) is only a little larger than a degree (1.5°, to be exact), and the other metric prefixes are also available: A millispry (mspry) is 1.11′. A microspry (μspry) is a truly small angle of rotation. Imagine the literary opportunity: “I called for him to stop, but he turned not a microspry.”

### The Big Question

Am I serious?

I don’t know. These musings are largely thought experiments because I don’t think that there’s a practical way to change all the various resources out there. The people who would need to change their ways are the ones least likely to see the need for change. I see things from the perspective of a high school teacher: What sort of presentation, in a fresh world without the baggage of hundreds of years, would be best for people learning this stuff for the first time?

According to Wikipedia, the concept of the radian is only three centuries old, and the term itself not even a century and a half. The degree is far, far older, but the spry is merely a multiple of the degree. There’s no reason why the spry couldn’t live alongside both the radian and the degree.

But convince the people who are set in their ways of that. Given the outrage that’s followed in the wake of the suggestion that 2π is a better constant than π itself, given the near-dogmatic worship of π as some sort of special, unique, miraculous number… it seems very daunting to introduce yet another competitor to that throne (even if the spry would not even unseat it).

*i* like it and’ll probably work up some intro-trig notes

based on units-of-fifteen-degrees if i ever again present

intro-trig and have the leeway even to try it out for one

lecture. but i wouldn’t dream of changing the textbook

vocabulary and notations for exams and such, of course.

here’s something i’ve done for years and maybe *should*

have done more publicly when i actually met actual students.

in many calculations there’s only one angle in play. “x”, say.

writing out “sin(x)” and “cos(x)” over and over

soon becomes tiresome. students spontaneously

write “sin” and “cos” in their place… which in context

isn’t a *bad* idea… but i’ve gone ’em one better and

when i’m working alone, write (greek letters not TeX code)

\sigma, \kappa, and \tau [for sin(x), cos(x), and tan(x)].

and my code’s much clearer for it.