angelcalc

1. Learning Assistance Center
Southern Maine Community College
ANGLE CALCULATIONS
An angle is formed by rotating a given ray about its endpoint to some terminal position. The
original ray is the initial side of the angle and the second ray is the terminal side of the angle.
The common endpoint is the vertex of the angle (see figure
below). The measure of an angle is determined by the amount
of rotation of the terminal ray from the initial ray.
For the purposes of this worksheet we will discuss two ways
to measure angles: by degrees and by radians.
vertex
initial side
Degrees
The concept of measuring angles in degrees grew out of the
belief of the early Babylonians that the seasons repeated every
360 days. One degree is the measure of an angle formed by
rotating a ray (one three hundred sixtieth) of a complete
revolution.
Figure 1
There are two popular methods for representing degrees and their fractional parts. One is the
decimal degree method. For example, the measure 29.76° is a decimal degree. It means
29° plus 76 hundredths of 1°
A second method of measurement is known as the DMS (Degree, Minute, Second) method. For
example, the measure 126° 12′ 27″ is a degree value expressed in DMS form. It means
126° plus 12 minutes plus 27 seconds
In the DMS method the fractional part of a degree may be expressed by understanding that we
subdivide a degree into 60 equal parts, each of which is called a minute (denoted by ′) and that a
minute is subdivided into 60 equal parts, each of which is called a second (denoted by ″). Thus
1º = 60′, 1′ = 60″, and 1º = 3600″.
Changing Minutes and Seconds to Decimal Degrees: It is sometimes necessary to change
minutes or seconds to decimal equivalents or vice versa. Minutes or seconds are first changed to
their fractional part of a degree. Then the fraction is changed to its decimal equivalent by
dividing the numerator by the denominator.
Remember : 1′ =
Anglecalc.doc
1
1
of a degree, and 1′′ =
60
3600
June 7, 2006

2. 2
Angle Calculations
• To change minutes to a decimal part of a degree: Divide minutes by 60.
• To change seconds to a decimal part of a degree: Divide seconds by 3600.
For example: Convert 50° 15′ 27″ to a decimal degree value.
15
60
27
Change 27″ to its decimal degree equivalent.
3600
= 0.25º
And then add the values together: 50°+ 0.25°+ 0.0075º
= 50.2575°
Change 15′ to its decimal degree equivalent .
= 0.0075°
Changing a Decimal Degree into a DMS Degree Value: The decimal part of a degree can be
changed to minutes and seconds by reversing the procedure. To change a decimal part of a
degree to minutes, multiply by 60. Similarly, to change the decimal part of a minute to seconds,
multiply by 60.
• To change a decimal part of a degree to minutes: Multiply the decimal part of a degree by 60.
• To change a decimal part of a minute to seconds: Multiply the decimal part of a minute by 60.
For example: Convert 50.75° into a DMS degree value
Change 0.75° to minutes
And so...
0.75 × 60
50.75°
= 45′
= 50° 45′
For example: Convert 28.43° into a DMS degree value
Change 0.43° to minutes and seconds 0.43 × 60
0.8 × 60
And so...
28.43°
= 25.8' (degrees to minutes)
= 48' (decimal part of min to sec)
= 28° 25' 48″
Adding and Subtracting Angle Measures: Angle measures can be added or subtracted. Keep
in mind that only like measures can be added or subtracted.
To add, arrange the measures in columns of like measures.
For example:
12° 15' 54"
+ 82° 28' 19"
94° 43' 73"
which simplifies* to 94° 44' 13″
*Since 73″ = 1' 13″, we must simplify and shift the values to the left as appropriate. In this case
we must add one minute to 43 to make 44 minutes leaving a remainder of 13 seconds.
Anglecalc.doc
June 7, 2006

3. 3
Angle Calculations Worksheet
To subtract, arrange the measures in columns of like measures; borrow as needed.
37° 15′ = 36° 75′ (Borrow 1° from 37°. 1° = 60′ and 60′ + 15′ = 75′.)
For example: - 15° 32′ = - 15° 32′
21° 43′
Since we can’t subtract 32 from 15 with real numbers, we must borrow (much like you would in
regular whole number subtraction) one degree to make more minutes from which to subtract.
Complementary & Supplementary Angles: If the sum of the measures of two angles equals
one straight line (180°), the angles are called supplementary. If the sum of the measures of two
angles equals one right angle (90°), the angles are called complementary. To find the
complement of any angle, subtract the angle from 90°; to find the supplement of any angle,
subtract the angle from 180°.
For example: Find the complement of 63° 37'.
90°
= 89° 60'
− 63° 37' = − 63° 37'
(Borrow 1° from 90°. 1° = 60′ )
26° 23'
Multiplying and Dividing Angle Measures: To multiply or divide angle measures, perform the
indicated operation and simplify as needed.
For example: An angle whose measure is 65° 02' 37" needs to be twice as large. Find the
measure of the new angle.
(65° 02' 37") * 2 = 130° 04' 74".
Since 74" = 1' 14", we must simplify to a final answer of 130° 05' 14".
PRACTICE: Perform the indicated operations. Be sure to simplify your final answers.
1. Change 0.42° to equivalent minutes and seconds.
Anglecalc.doc
June 7, 2006

4. 4
Angle Calculations Worksheet
2. Change 15° 4' to its decimal degree equivalent rounded to the nearest ten-thousandth.
3. Change 0.46° to equivalent minutes and seconds.
4. Change 8° 20' to its decimal degree equivalent rounded to the nearest ten-thousandth.
5) Add and simplify:
7) Add and simplify:
15° 47' 18"
+ 37° 12' 45"
45° 10' 14"
+ 7° 8' 55"
6) Subtract and simplify:
147° 28'
− 114° 35' 23"
32°
6"
8) Subtract and simplify: − 20° 10' 8"
9. Find the measure of an angle with a complement of 35°.
Anglecalc.doc
June 7, 2006

5. 5
Angle Calculations Worksheet
10. Find the measure of an angle with a supplement of 35°.
11. An angle whose measure is 17° 36' 40" needs to be three times as large. Find the
measure of the new angle in degrees and minutes.
12. An angle whose measure is 45° 37' 30" needs to be twice as large. Find the measure of.
the new angle in degrees and minutes.
13. A right angle will be divided into four equal angles. Find the measure of each new angle in
degrees and minutes.
14. Find the complement of 40° 37' 26". Then convert the result to its decimal equivalent
rounded to the nearest ten-thousandth.
ANSWER KEY
1. 25' 12"
8. 11° 49' 58"
Anglecalc.doc
2. 15.0667°
9. 55°
3. 27' 36"
10. 145°
4. 8.3333°
11. 52° 50'
5. 53° 3"
12. 91° 15'
6. 32° 52' 37"
13. 22° 30'
7. 52° 19' 9"
14. 49.3761°
June 7, 2006

6. 6
Angle Calculations Worksheet
Radians
Until now, we have been measuring angles in terms of
degrees, minutes and seconds, or in degrees and decimal
parts of degrees. In order to deal with sinusoidal
waveforms and some other important ideas of physics and
calculus, another system of angle measurement is
necessary, namely, radians.
1 radian
(57.3 degrees)
radius = x
Technically, at one radian the measure of the radius (half
the distance across any particular circle) is equal to the arc
length marked by the intersection of the two rays of the
angle on any particular circle (see figure at right).
So what’s the relationship between degrees and radians? Study the figure to the left. For every
angle measured in degrees, there is a related radian measure. The values on the inner ring are
radian measurements. For
example, for every 180 degrees
there are π radians. Therefore,
180° = π radians. We can use
this understanding of the
relationship between radians and
degrees to go from one to the
other using the process of
dimensional analysis or unit
conversion.
While it is possible to use your
calculator to transpose radians
into degrees and degrees into
radians, it is good practice and
probably faster and easier to do it
with some quick hand
calculations as demonstrated
next.
Anglecalc.doc
June 7, 2006

7. 7
Angle Calculations Worksheet
For example: 120° is equal to how many radians?
2
1 2 0° π rad 2π
/ / //
120° =
×
=
rad
1
3
18 0 °
// //
3
For example: 3π radians equal how many degrees?
3π =
3π rad 180° 540°
/ // /
×
=
= 540°
1
1
π rad
/ // /
PRACTICE: Perform the indicated operations. Be sure to simplify your final answers.
Convert the following degree measures to radians:
1) 20°
2) 50°
3) 270°
Convert the following radian measures to degrees:
4) 2 radians
Answers: 1)
π
90
Anglecalc.doc
5)
radians
2)
5π
radians
18
π
2
3)
6) 0 .25 radians
radians
3π
radians
2
4)
115°
5)
90°
6)
14°
June 7, 2006