Lesson Objectives
• Learn how to work with degrees, minutes, and seconds
• Learn how to convert between decimal degrees and degrees, minutes, and seconds

Calculating with Degrees, Minutes, and Seconds

We previously learned that a complete rotation of a ray gives an angle measure that is 360° When we make 1/360 of a full rotation, the angle measure is 1°. In some cases, we will need to work with portions of a degree. In this case, we can use minutes and seconds. One minute, which is written as 1' is 1/60 of a degree:
1' = 1°/60
60' = 1°
Additionally, one second, which is written as 1", is 1/60 of a minute or 1/3600 of a degree:
1" = 1'/60 = 1°/3600
60" = 1'
Let's look at an example.
Example #1: Perform each calculation.
31° 18' + 15° 43'
To perform the calculation, we add the degrees and minutes separately:
31° + 15° = 46°
18' + 43' = 61'
61' = 1° 1'
We have 46 degrees and 61 minutes. Since 60 minutes equals one degree, we can convert 61 minutes into 1 degree and 1 minute to give us a final answer of 47 degrees and 1 minute:
47° 1'
Example #2: Find the supplement of the given angle.
113° 15' 29"
We know that supplementary angles are two positive angles whose sum is 180°. We can subtract 180° minus our given angle measure to obtain the supplement of the angle.
180° - 113° 15' 29"
To perform the calculation, we need to rewrite things and do a little borrowing:
180° 0' 0" - 113° 15' 29"
Let's take 1 degree away from 180° and change it into 60':
179° 60' 0" - 113° 15' 29"
Let's take 1 minute away from 60' and change that into 60":
179° 59' 60" - 113° 15' 29"
Now we can subtract:
60" - 29" = 31"
59' - 15' = 44'
179° - 113° = 66°
Putting everything together, we obtain the supplement as:
66° 44' 31"

Converting between Decimal Degrees and Degrees, Minutes, and Seconds

In some cases, we will need to convert between decimal degrees and degrees, minutes, and seconds. Let's look at some examples.
Example #3: Convert to decimal notation.
13° 15' 45"
We take the degree part of 13 and place it in front of our decimal point.
13.
For the part after the decimal point, we need to divide the minutes by 60 and the seconds by 3600. The sum of these two decimals will be our decimal part.
15/60 = 0.25
45/3600 = 0.0125
0.25 + 0.0125 = 0.2625
13° 15' 45" = 13.2625
Example #4: Convert to degrees, minutes, and seconds.
111.375°
We leave the whole number part of 111 alone and just work with the decimal part of .375.
Multiply .375 by 60 to convert to minutes.
.375 • 60 = 22.5
Since we have 22.5', we can keep the whole number part of 22, and work with the decimal part of .5.
Multiply .5 by 60 to convert to seconds.
.5 • 60 = 30
111.375° = 111° 22' 30"

Skills Check:

Example #1

Find the complement of the angle given.

39° 15' 37"

A
50° 44' 23"
B
35° 18' 19"
C
140° 44' 21"
D
11° 15' 27"
E
37° 15' 39"

Example #2

Convert to DMS.

88.515°

A
88° 4' 9"
B
89° 1' 7"
C
54° 30' 88"
D
88° 19' 17"
E
88° 30' 54"

Example #3

Convert to decimal degrees.

22° 30' 9"

A
22.475°
B
22.156°
C
22.5025°
D
22.313°
E
22.583°