Revolutions

We seldom learn to use the simplest, most organic unit of measure for geometric angles, the revolution (rev). Other names because that this unit room full circle, turn, full turn, and also rotation (rot). These are all good names and also they all mean the exact same thing.

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1 full circle = 1 rev = 1 revolve = 1 rot = 360°

degrees

In elementary school, we discover that angles space measured in degrees (°).

1 complete circle = 360°

Historical note: The number 360 comes down to us from antiquity. 360 was useful to old astronomers because it is about the variety of days in the year. Also, that is useful because it is divisible through 2, 3, 4, 5, 6, 8, and also 10. But the number 360 is arbitrary, not fundamental. If the ancients had characterized the full circle to be some other variety of degrees, climate we"d be using that number today.

In high school trigonometry and calculus classes, we learn that mathematicians favor radians (rad).

1 full circle = 2π rad

Note: If you have actually not yet learned around radians in school, girlfriend may overlook the radians in every little thing below.

Historical note: The radian was designed in the 1700s through mathematicians who wanted to define angles rationally, no using any kind of arbitrary numbers like 360. They can have rationally defined the complete circle to it is in 1, yet instead they identified the full circle to it is in 2π, the one of a one of radius 1. This definition simplified countless equations, especially equations in ~ the deepest levels of mathematics, by removing factors of 2π. However, it complicated many other equations, particularly equations involving basic geometry, through introducing determinants of 2π. The main trouble with radians is that the base unit, one radian, is awkward: 1 radian = 180/π° = 57.2958°. This is perplexing and useless for normal people. Nobody ever builds noþeles that has actually an edge of 1 radian, 2 radians, or any type of integer variety of radians. Unfortunately, mathematicians have actually deep reasons for maintaining radians, therefore we are stuck with them.

## Comparing Revolutions, Degrees, and Radians

Let"s compare revolutions and degrees (and radians).

no turn 0 0
quarter turn 1/4 90° π/2
half turn 1/2 180° π
three-quarter turn 3/4 270° 3π/2
full turn 1 360°
twelfth turn 1/12 30° π/6
eighth turn 1/8 45° π/4
sixth turn 1/6 60° π/3
fifth turn 1/5 72° 2π/5
third turn 1/3 120° 2π/3
two turns 2 720°
three turns 3 1080°

To convert from transformations to degrees, multiply by 360. To convert from levels to revolutions, division by 360.

When friend use levels you are frequently working through integers, but when you use revolutions (or radians) you are frequently working through fractions (or decimals). Hand calculations are sometimes simpler when you use revolutions yet sometimes less complicated when you usage degrees. It"s good to recognize both ways.

Revolutions (turns) space a more rational and also natural unit the measure than degrees. You"ll gain a deeper expertise of angle if friend think about revolutions quite than degrees. An angle is an ext fundamentally a subdivision the a circle fairly than a amount of degrees. Because that example, a best angle is much more fundamentally a 4 minutes 1 of a circle rather than a sum of 90 degrees.

Let"s divide the circle right into n equal sectors (see diagram below). The angle of every sector is 1/n rev = 360/n° = 2π/n rad. It is easier to recognize this if friend think about revolutions quite than degrees (or radians).

Let"s look in ~ some basic geometry using revolutions and also degrees (and radians). The diagram below shows supplementary angles, safety angles, and also triangles. The ideas are clearer if you think around revolutions rather than degrees. The arithmetic may be simpler using degrees if you have actually trouble adding and individually fractions.

Let"s look at at polygons (see diagram below). For a continual polygon v n sides, the exterior angle is 1/n rev = 360/n° = 2π/n rad. That is less complicated to understand this if you think about revolutions quite than degrees (or radians). The interior angle is the supplement of the exterior angle.

## Teaching Revolutions, Degrees, and also Radians

I think it would be great if teacher would introduce revolutions (turns) in ~ the very same time the they introduce degrees. This will assist the students to know angles at a more fundamental level, much less dependent top top the arbitrarily magic number 360. Teachers already introduce the general concept of transformations (turns) as soon as they speak things like "a complete circle is 360°", yet they have the right to make the concept an ext numerically an exact by saying "a complete turn is 360°, a fifty percent turn is 180°, a quarter rotate is 90°, and an eighth turn is 45°" or writing "1 rev = 360°, 1/2 rev = 180°, 1/4 rev = 90°, and also 1/8 rev = 45°". Students should occasionally practice doing a few calculations using changes (turns) rather than degrees. The course, student will must spend most of their time finding out to calculation with levels (and later, radians), since that is the standard.

## Angles in Trigonometry and also Calculus

Finally, let"s take a rapid look at much more advanced mathematics: trigonometry and also calculus.

We can take into consideration using transformations with trigonometric features (sine, cosine, tangent). Because that example, rather of saying cos(60°) = 1/2 or cos(π/3) = 1/2 utilizing radians, we can want come say cos(1/6) = 1/2 utilizing revolutions. However this is no practical because we count on calculators to evaluate the trigonometric functions, and also calculators generally have only DEG and also RAD modes, no REV mode.

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When we go to deeper levels of mathematics, such together calculus and also mathematical analysis, it transforms out the radians room the most rational and natural units. Because that example, consider this basic equation: the limit of sin(x)/x together x approaches 0 is exactly 1. This equation would certainly not be so elegant if us used any unit other than radians.