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Radians and Degrees (page 1 of 2)

Sections: Radians and Degrees, Reducing Angles to Useful Values

Radians and degrees are two units for measuring angles. There are at least four such units, but degrees and radians are the ones you are most likely to encounter in high school and college.


Degrees are used to express directionality and angle size. If you stand facing directly north, you are facing the direction of zero degrees, written as 0. If you turn yourself fully around, so you end up facing north again, you have "turned through" 360; that is, one revolution (one circle) is 360.

Aside: Why is one revolution divided into 360 parts called "degrees"? Because the ancient Babylonians, now four- or five-thousand years dead, viewed the numbers 6, 12, and 60 as having particular religious significance. It is because of them that we have twelve-hour nights and twelve-hour days, with each hour divided into sixty minutes and each minute divided into sixty seconds. And "once around" is divided into 660 = 360 parts called "degrees".

So a full revolution is 360 and a half-turn (an "about face") is 180. If you start by facing north again and then turn to the east, you will have made a 90, or one-quarter, turn, and you will be facing toward 90. If you start facing north and then turn to the west, you will have made a 90 turn, but you will be facing toward 270. This is because directional degrees start at 0 for "north" and then go around clockwise.




If, when making your one-quarter turn from "north" to "west", you held your arm straight out in front of you, your arm would be said to have "swept out" a 90 angle. This angle would have been formed by your arm's starting position (the "initial" side of the angle) and your arm's ending position (the "terminal" side of the angle). The path of your fingertips as your arm moved would be an "arc" and the angle you turned through is said to "subtend" that arc.

Note: When directions are given in terms of degrees, the direction is found by starting at "north", being 0, and moving clockwise by the number of degrees given. Another way of giving directions using degress is of the form N36W or S27E. These mean "36 degrees west of north" and "27 degrees east of south", respectively. Whatever convention your book uses should be defined in the book; ask your instructor, if it isn't otherwise clear.

And yes, this is different from how you'll be measuring angles. Then you'll start with 0 being "east", sort of, and you'll rotate anti-clockwise. Copyright Elizabeth Stapel 2010-2011 All Rights Reserved

Decimal Degrees and DMS

When you work with degrees, you'll almost always be working with decimal degrees; that is, with degrees expressed as decimal numbers such as 43.1025. But just as "1.75" hours can be expressed as "1 hour and 45 minutes", so also "degrees" can be expressed in terms of smaller units. These units, just as for "hours", are called "minutes" and "seconds". Just as "hours" can be expressed as decimals or else as hours - minutes - seconds, so also "degrees" can be expressed as decimals or else as degrees - minutes - seconds, denoted as "DMS".

  • Convert 43.1025 to DMS form.
  • I can see that I have 43, but what do I do with the "0.1025" part? I treat it like a percentage of the sixty minutes in one degree, and find out how many minutes this is:

      (0.1025 degrees)(60 minutes / 1 degree) = 6.15 minutes

    ...or 6 minutes and 0.15 of a minute. Each minute has sixty seconds, so:

      (0.15 minutes)(60 seconds / 1 minute) = 9 seconds

    Then 43.1025 = 43 6' 9"

Notice the symbols: A single quote-mark (an apostrophe) indicates "minutes" and a double quote-mark indicates "seconds". This is similar to the notation (in Imperial measurements) for "feet" and "inches": the smaller unit gets the more-substantial mark.

  • Convert 102 45' 54" to decimal form.
  • Clearly, I've got 102, but how do I convert the minutes and seconds to decimal form? By using the definitions and doing the divisions. The 45' means 45/60 of a degree, since each degree contains sixty minutes. Simplification and long division gives me 45/60 = 3/4 = 0.75. So the 45' is 0.75.

    Now I need to deal with the 54". Since each minute is sixty seconds, then I get 54/60 = 9/10 = 0.9. But this is minutes. Now I need to convert the 0.9 of a minute to degrees:

      (0.9 minutes)(1 degree / 60 minutes) = 0.015 degrees

    So 102 45' 54" = 102 + 0.75 + 0.015 = 102.765.


Why do we have to learn radians, when we already have perfectly good degrees? Because degrees, technically speaking, are not actually numbers, and we can only do math with numbers. This is somewhat similar to the difference between decimals and percentages. Yes, "83%" has a clear meaning, but to do mathematical computations, you first must convert to the equivalent decimal form, 0.83. Something similar is going on here (which will make more sense as you progress further into calculus, etc).

The 360 for one revolution ("once around") is messy enough. Why is the value for one revolution in radians the irrational value ? Because this value makes the math work out right. You know that the circumference C of a circle with radius r is given by C = 2πr. If r = 1, then C = 2π. For reasons you'll learn later, mathematicians like to work with the "unit" circle, being the circle with r = 1. For the math to make sense, the "numerical" value corresponding to 360 needed to be defined as (that is, needed to be invented having the property of) "2π is the numerical value of 'once around'."

Converting Between Radians and Degrees

Each of radians and degrees has its place. If you're describing directions to me, I'd really rather you said, "Turn sixty degrees to the right when you pass the orange mailbox", rather than, "Turn one-third π radians" at that point. but if I need to find the area of a sector of a circle, I'd rather you gave me the numerical radian measure that I can plug directly into the formula, rather than the degree measure that I'd have to convert first.

But you won't always be given angle measures in the form you'd prefer, so you'll need to be able to convert between radians and degrees. To do this, you'll use the fact that 360 is "once around", and so is 2π. However, you'll use this fact in the form of the somewhat simplified correspondence of 180 to π.

  • Convert 270 to radians.
  • Since 180 equates to π, then:

      (270/1)*(pi/180) = (3/1)*(pi/2) = (3pi)/2

    The equivalent angle is  (3pi)/2

  • Convert  pi/6  radians to degrees.
    • (pi/6)*(180/pi) = (1/1)*(30/1) = 30

    The equivalent angle is 30

Note that the way I used the correspondence varied with what I was given. If I needed to end up with radians, I put π on top; if I needed to end up with degrees, I put 180 on top. That's all there is to that.

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Cite this article as:

Stapel, Elizabeth. "Radians and Degrees." Purplemath. Available from Accessed


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