Measuring Angles and Rotations

UNH Mathematics Center

Degree measure is customary.
Radian measure is convenient.

1  Degree measure

A newcomer to our world might wonder why the number 360 shows up so prominently in this measurement system. Perhaps it's been that way too long for us to give a complete reason now. We do know that the Babylonians long ago counted by 60's. (It's even possible that there's a connection with the number of days in a year. Calendars weren't always what they are now.)

Longitude lines on maps are still marked in degrees: they are numbered up to 180° east, and 180° west, of the ''prime meridian'' (longitude 0°) that passes through the observatory at Greenwich, England.

We have to say that 360 has been a customary ``important number'' for a long time.

When we measure angles and rotations in degrees,

1. there are 360° in one complete revolution (turn once around in a circle, to face in the original direction at completion)
2. to reverse our direction, we turn one-half revolution or 180° (which is why people will complain that you've done a ''180° change'' if you change your mind)
3. a right angle, whose angle sides are perpendicular, measures one-quarter revolution or 90° (90° is the ''right'' angle for a housebuilder to use)
4. A really good way to picture rotations is to remember that one-twelfth of a complete revolution measures 360°/12 = 30°. The acute angle between ''2 o'clock'' and ''3 o'clock'' is 30°; the one between ''10:00'' and ''noon'' is 60°.

2  Radian measure

The number p often shows up in radian measurements because of the natural connection between the angle measurement and a circle's circumference. You don't have to use the number p, explicitly, when writing the radian measure of an angle. A particular angle's radian measurement is a fixed number and can be expressed (or approximated) in decimals. Remember that p is approximately 3.14159.

When we measure angles and rotations in radians:

1. There are 2p radians in one complete revolution. (Turn once around in a circle, to face in the original direction at completion.)
2. To reverse our direction, we would turn one-half revolution, or p radians.
3. A right angle, whose angle sides are perpendicular, measures one-quarter revolution or p/2 radians (p/2 is the ''right'' angle for a housebuilder to use).
4. You can picture rotations by remembering that one-twelfth of a complete revolution measures 2p/12 = p/6 radians. For instance, the acute angle between ''2 o'clock'' and ''3 o'clock'' is p/6 radians; the one between ''10:00'' and ''noon'' is p/3 radians.

3  Rotational directions, and large rotations

If we take the viewpoint of rotations we can see that an endless variety of rotations might begin at 3 o'clock and end at 2 o'clock:

Some might be more than one complete rotation, for example We would measure this rotation as either 390°, or 13p/6 = 2p+ p/6 radians.

And some might be in the other direction: The ''other direction'' - clockwise - is taken to be the negative direction. This rotation measures either -336°, or -11p/6 radians.

Here are some other rotations that are all related to a ''two clock-hours'' angle, but are different rotations:

• 7p/3 radians, or 420°:
• p/3 radians, or 60°:
• -p/3 radians, or -60°:

Here are some things you should notice about rotations:

• Notice that counterclockwise is the ''positive'' direction for rotations, and clockwise is the ''negative'' direction.

• You can see we can assign either a degree or a radian measure to any rotation, no matter how large or small, and no matter which direction the rotation has.

• Having the same initial and final sides does not mean that two rotations are the same!

4  Choosing a measure

Degree measure is used in most ''traditional'' trigonometric measurements of the ground. Surveyors will usually use degree measure. Degree measure is also the one usually associated with describing directions by bearings.

Radian measure is the system you must use for calculus problems! The reason is that the formulas for the derivatives of trigonometric functions are all derived assuming that radian measures are used. (This makes the formulas much simpler than they would otherwise have been.)

If you try to use calculus with the trig functions, but use degree measure instead of radian measure, your numerical answers will come out wrong. For instance: even though ''p radians'' and ''180 degrees'' are corresponding measures for the same rotation, as numbers they are different. 3.14159 is simply not the same as 180.

4.1  Converting between the two systems

• To convert radians to degrees, multiply by 180/p.
• To convert degrees to radians, multiply by p/180.
• If you don't want to spend your life doing arithmetic, learn to picture the rotations instead, and ''read them off'' in either measure. This is by far the easier approach. It's no more difficult than learning to tell time from a clock with Roman numerals.
  • One whole revolution is 2p radians, or 360°.
  • One right angle is p/2 radians, or 90°.
  • One ''clockface hour,'' one-twelfth of a whole revolution, is alternately p/6 radians, or 30°.