Angular Units of Measure
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Skills Application: Angular Units of Measure
Step-by-Step: Angular Units of Measure
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One angular unit of measure was developed by the ancient Babylonians 4000 years ago. It featured multiples of the number 60. We still use it today, this is why we have 360 ˚ in a full circle (6 x 60), 180 ˚ is a straight line (3 x 60) and 90˚ in a right angle (6 x 60 ÷ 4) and so on.


The measuring of time that we use today also evolved out of this ancient Babylonian 60-based system: 12 months make a year (12 = 60 ÷ 5), 24 hours a day (60 ÷ 5 x 2), 60 minutes an hour and 60 seconds a minute.

Angular Units
of Measure

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In most practical work, angles are measured in degrees and fractions of a degree. Smaller units have been defined as follows:

60 minutes = 1 degree         abbreviated as 60′ = 1°

60 seconds = 1 minute        abbreviated as 60′′ = 1′

In some trades, angles are usually rounded to the nearest degree. For most technical purposes, angles can be rounded to the nearest minute. For a few trades, angles are rounded to 2 decimals of a second.

Check out the ADVANCED tab. It is very important to read about degrees and other units of angular measure and use a modern scientific calculator. Get a $20-or-so dual-display scientific calculator, even if you can`t take it with you into the test hall. A calculator is also a learning tool, Technology Use is one of the 9 Essential Skills you can`t do without. Trying out the angular functions on a calculator will further your understanding and help with building a bigger mental picture. It builds links between numbers and algebra. It is important to try alternative calculations or approaches, to sketch & label angles, tracking changes to units of measure, visualizing to cross-link the concepts of angles and numbers. If you spend 20 – 40 min working with the different ideas in the ADVANCED page, you`ll remember key information easier and you`ll learn new skills faster.


A Short Review

The DMS system (degrees-miutes-seconds) of angular units of measure:

1 degree = 60 arcminutes

1 minute = 60 arcseconds

1 degree = 3600 arcseconds

1 full circle = 360 degrees = 21,600 arcminutes = 1,296,000 arcseconds

Scientific calculators come with DMS capability, read your manual.

Degrees can be added or subtracted from each other.

Degrees can be multiplied or divided by a whole number, fraction or decimal, but not degrees.

GetMath Worksheet Level1 novertspaac

Worksheet: Level 1

The operations used are clearly specified. Only one type of mathematical operation is used in a task.

Worksheet: Level 1 Sample Questions


  1. How many degrees make a full circle?

Further examples:

What is the conversion factor between:

  1. arcminutes (angular minutes) and degrees?
  2. arcseconds (angular seconds) and arcminutes (angular minutes)?
  3. degrees and arcseconds?

Worksheet: Level 1 Answer Key


  1. 360

Download Worksheet: Level 1

GetMath Worksheet Level2 novertspaac

Worksheet: Level 2

Tasks involve one or two types of mathematical operation. Few steps of calculation are required.

Worksheet: Level 2 Sample Questions


True or False?

  1. Degrees and angles is a 2-D, flat concept with no relevance to 3-D real objects, such as walls, corners, counter tops.

Further examples:

  1. Can navigators be a few arcseconds off their course without this causing problems?
  2. Can machined surfaces be out-of-alignment by a few arcminutes? Why or why not?

Worksheet: Level 2 Sample Answer Key


  1. False, angles relate to the 3-D real world.

Download Worksheet: Level 2

GetMath Worksheet Level3 novertspaac

Worksheet: Level 3

Tasks require a combination of operations. Several steps of calculation are required.

Worksheet: Level 3 Sample Questions


Calculate using a scientific calculator with a DMS button:

  1. 95° + 28° =
  2. 259° − 128° =
  3. 52° × 3 =
  4. 182° ÷ 6 =

Further examples:

  1. 95° + 3” =
  2. 5° – 3” =
  3. 15.32° x 4 =
  4. 120° ÷ 2.1 =

Worksheet: Level 3 Sample Answer Key


  1. 123°0'0"
  2. 131°0'0"
  3. 156°0'0"
  4. 30°20'0"

Download Worksheet: Level 3

GetMath Worksheet Additional novertspac

Infosheet: Advanced Additional Information

Tasks involve multiple steps of calculation. Advanced mathematical techniques may be required.

Additional Information

 Angles can be 2-D or 2 dimensional or 3-D or 3 dimensional.

Angles are indicated by 2 short lines that joint at a vertex:   Measured angles are indicated with a short arc line between the rays, lines, line segments that form the angle:angle measured icon  There are many synonyms used to indicate flat or spatial angles, see table below.

An angle is also the size of the opening between 2 lines, or the measure of rotation in circles. Rotation from a reference line can be clockwise or counter-clockwise. Angles can be positive and negative with respect to a reference line, for example “4° below the horizon” is a negative angle. There often are negative rake angles on plastic or metal cutting saw teeth or cutter head blades.

Angles can be measured using a number of units of measure, none of which are originating from either metric or imperial systems of measure. Some units include:

  • degrees, arcminutes, arcseconds (a.k.a. DMS, 360 parts make a full circle)
  • gradians (400 parts make a full circle)
  • radians (2π parts make a full circle)
  • angular mils (6000, 6300 or 6400 parts to make a full circle, depending on country)
  • compass points (32 or 64 parts make a full circle)
  • fractions or percent can be used to express an angle (three-quarter turn, 2% slope)

Many of these units are used only in specific industries. For example the 6400-part measure is used by NATO in military context. Radians are used in algebra and engineering. Scientific calculators can be set to operate in some of these completely different units of measure. Make sure your calculator is set to the system you are working in. Check your user manual how to change your calculator`s settings.

Here is a table to show some of the differences and equivalencies among some angular units of measure:

Amount of turn
° degrees, ` minutes, “ seconds
Full circle
2p rad
400 gon
Half circle
p rad
200 gon
Quarter circle
p/2 rad
100 gon
Eight circle
p/4 rad
50 gon
Twelfth circle
p/6 rad
33  1/3 gon
Sixteenth circle
22.5° or 22° 30`
p/8 rad
25 gon
1 degree
p/180 rad
1  1/9 gon
1 radian
57.295° or 57° 17` 44.81”
1 rad
636.619 gon
1 gradian
0.9° or 54`
p/200 rad
1         gon


Find out:

  • which buttons toggle between these units (DEG-RAD-GRAD) on your calculator, and
  • try the calculations mentioned in the video (30°+50°= 80°, etc.) with your calculator set to each of the 3 of the angular units. Do you get the same results in RAD and GRAD modes as in DEG mode? Find out how your calculator works, different models are programmed differently.

Some other angular units of measure are funny or anecdotal, they work unofficially as figures of speech. “A quarter turn” or a “U-turn” is understood clearly widely without using any numbers, so is “give it 2 cranks” or “9 o`clock position”.

“When the Sun/Moon is at that treetop” relates to an angle less precisely, so does “head for the hills” or “follow the road sign” or “give it more air”. These could still be meaningfully used to indicate angles well enough during rescue or evacuation. Viewing angles of screens and TVs can be referred to as well without using numbers: “… squish down that afro fuzz, I can`t see the left side …” Sight lines or blind spots in traffic have to do with angles again without numbers: “I coulnd`t see the speed limit sign, officer, Grandma`s hat was in the way”.

Property of matter being measured:
possible synonyms
Unit of measure
adopted into Metric, Imperial & US Customary
turn, cycle, revolution, rake, bevel, miter, clearance, relief, hook
, angle measured icon
degrees-minutes-second or DMS
° ` “


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