Hamilton's Math To Build On - copyright 1993

Degrees

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    Degrees are the units of measurement for angles.

    There are 360 degrees in any circle, and one
    degree is equal to 1/360 of the complete
    rotation of a circle.

    360 may seem to be an unusual number to use, but this part
    of math was developed in the ancient Middle East. During
    that era, the calendar was based on 360 days in a year, and
    one degree was equal to one day.


* Fractions of Degrees

There are two methods of expressing fractions of degrees.
  1. The first method divides each degree into 60 minutes (1° = 60'), then each minute into 60 seconds (1' = 60").

    For example, you may see the degrees of an angle stated like this: 37° 42' 17"

    The symbol for degrees is ° , for minutes is ', and for seconds is ".

  2. The second method states the fraction as a decimal of a degree. This is the method we will use.

    An example is 37° 42' 17" expressed as 37.7047° .

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Most scientific calculators can display degrees both ways. The key for degrees on my calculator looks like ° ' ", but the key on another brand may look like DMS. You will need to refer to your calculator manual to determine the correct keys for degrees. Most calculators display answers in the form of degrees and a decimal of a degree.

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It is seldom necessary to convert from minutes and seconds to decimals or vice versa; however, if you use the function tables of many trade manuals, it is necessary. Some tables show the fractions of degrees in minutes and seconds (DMS) rather than decimals (DD). In order to calculate using the different function tables, you must be able to convert the fractions to either format.

* Converting Degrees, Minutes, & Seconds to Degrees & Decimals

To convert degrees, minutes, and seconds (DMS) to degrees and decimals of a degree (DD):

  1. First: Convert the seconds to a fraction.
    Since there are 60 seconds in each minute, 37° 42' 17" can be expressed as
    37° 42 17/60'. Convert to 37° 42.2833'.

  2. Second: Convert the minutes to a fraction.
    Since there are 60 minutes in each degree, 37° 42.2833' can be expressed as
    37 42.2833/60° . Convert to 37.7047° .


Degree practice 1: Convert these DMS to the DD form. Round off to four decimal places.

     (1) 89°  11' 15"        (5) 42°  24' 53"
     (2) 12°  15'  0"        (6) 38°  42' 25"
     (3) 33°  30'            (7) 29°  30' 30"
     (4) 71°   0' 30"        (8)  0°  49' 49" 
Answers.

* Converting Degrees & Decimals to Degrees, Minutes, & Seconds

To convert degrees and decimals of degrees (DD) to degrees, minutes, and seconds (DMS), referse the previous process.

  1. First: Subtract the whole degrees. Convert the fraction to minutes. Multiply the decimal of a degree by 60 (the number of minutes in a degree). The whole number of the answer is the whole minutes.

  2. Second: Subtract the whole minutes from the answer.

  3. Third: Convert the decimal number remaining (from minutes) to seconds. Multiply the decimal by 60 (the number of seconds in a minute). The whole number of the answer is the whole seconds.

  4. Fourth: If there is a decimal remaining, write that down as the decimal of a second.

       Example: Convert 5.23456°  to DMS.

       5.23456°  - 5°  = 023456°                 5°  is the whole degrees
       0.23456°  x 60' per degree = 14.0736'   14 is the whole minutes
       0.0736' x 60" per minutes = 4.416"     4.416" is the seconds
    DMS is stated as 5° 14' 4.416"


Degree practice 2: Convert these DD to the DMS form.

     (1) 75.25°              (5) 13.12345° 
     (2) 43.375°             (6) 21.5° 
     (3) 9.5625°             (7) 59.7892° 
     (4) 33.9645°            (8) 65.1836° 
Answers.

On to Degrees: Circles, Triangles, Rectangles, Squares

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