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Angles and Unit circle – Angles, Angular units and Converting angles from one unit to another – pg.1/7

Posted in Math Theory by Lia on 05/01/2017

Angles and Unit circle: pg. 1, 23, 4, 5, 6, 7



pg.1…..Angles and Unit circle  – Angles, Angular units and Converting from one angular unit to another. 

pg.2…..Angles and Unit circle  – Unit circle

pg.3…..Angles and Unit circle  – Types of angles and Angle pairs

pg.4…..Angles and Unit circle  –  Standard position of an angle and Reference angles

pg.5…..Angles and Unit circle  – Coterminal angles

pg.6…..Angles and Unit circle – exercises

pg.7…..Angles and Unit circle – answers to exercises


1. Angles

Angle, is a geometrical figure that has two semi-lines (named rays or sides of the angle) with same origin (named vertex), as shown below.


The angle symbol is “” . Some times this symbol is omitted.

It is very common that the angles are denoted by Greek letters like α (alpha), β (beta), γ (gamma) and so on.

In geometry, some times, the angles can be identified with the letter that represents the vertex of the angle or, with three letters that are representing  the three points that form the angle’s sides (including the vertex). When the angle is identify by three letters, the middle letter is the letter that identifies the angle’s vertex.



2. Angular units

There are several units to measure the angels. From those units, the degrees and radians are the most used.


2.1 Degrees (arc-degrees)

The symbol is: ” ° ” (degree) .

A full circle has 360° (360 degrees).


The degree can be divided further as shown bellow: 

1° = 60′ (minutes or arc-minutes)

1′ = 60″ (seconds or arc-seconds)


2.2 Radians 

The symbol is: ” rad ” (radian) .

A full circle has 2π radians.

 π (read pi) is a constant (an irrational number) that is approximate  3.14159…. radians. In most calculations, π is consider to be equal with 3.14 radians.  

An angle measured in radians may or may not have in its value the symbol π. 


Usually when the value of the angle contains the  constant π in it, the symbol “rad” can be omitted.

The radian can be divided further as shown bellow:

1 rad = 1000 mrad (or milliradians)

1 mil = 1000 μrad (or microradians)


2.3 Some other angular units

  • Cycle (turn)

         No special symbol.

         A full circle has 1 cycle (turn).

  • Gon (grade or gradian)

          The symbol is: “” (gon).

          A full circle has  (gons).

  • and so on.


On this post all angles will be measured only in degrees and/or radians.


3. Converting from one angular unit to another. 

(degrees into radians and radians into degrees)


As mention above, a complete circle has an angle of 360° or radians. This means that 360° is equivalent with 2π radians, therefore 180° is equivalent with π radians (the results of dividing 360° and 2π by 2).

This means that, if 180° is equivalent with π,  an angle α measured in degrees will be equivalent with {{{pi}{{alpha}^{circ}}}/{180^{circ}}} radians.

Reverse, if  π is equivalent with 180°, an angle α measured in radians will be equivalent with {{{180^{circ}}{alpha}(rad)}/{pi}} in degrees.




If π radians is replaced with its numerical value (approx. 3,14… radians), than the value of the angle will be a numerical value. By using π instead of its numerical value, the angle measurement is more precise.


1. Examples of converting from one angular unit to another: 

2. Examples of transforming an angle measured in radians that has in its value the symbol π, into an angle measured in radians that does not has in its value the symbol π (the result will be a numerical value).




Angles and Unit circle: pg. 1, 2,  3, 4, 5, 6, 7







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