UNITS OF MEASUREMENTS OF ANGLES

UNITS OF MEASUREMENTS OF ANGLES
i) Sexagesimal system or British system : The Sexagesimal system is the most prevalent system of measurement where a right angle is divided into 90 equal parts called Degrees. Each degree is divided into 60 equal parts called Minutes, and each minute into 60 equal parts called Seconds.
    The symbols \({1^0},{1^1}and{1^{11}}\) are used to denote a degree, a minute and a second respectively
In short, 
    1 right angle =90⁰ (or 90 degrees)
    1⁰(or 1 degree) =60¹ (or 60 minutes)
    1¹(or 1 minute) =60¹¹ (or 60 seconds)

ii) Centesimal system or French System :  In the Centesimal system, the right angle is divided into 100 equal parts, called Grades ; each grade is subdivided into 100 minutes, and each minute is subdivided into 100 seconds.
The symbol is used to denote a grade.
In short, 
  90⁰  (or 1 right angle) =100^g (100 grades)
  1^g  (or 1 grade) =100¹ (or 100 minutes)
  1¹  (or 1 minute) =100¹¹ (or 100 seconds)
Note : It is clear that minutes and seconds in sexagesimal and centesimal systems are different
iii) Circular System or Radian Measure : In the circular system, the radian measure of an angle is introduced using arc lengths in a circle of radius r.
Radian : The angle subtended from the centre of a circle which intercepts an arc equal in length to the radius of the circle.
    It is represented by rad or ^c.
Ex : 1.5 radians is written as 1.5 rad or 1.5^c
The angle subtended by an arc in radians of a circle is defined as the ratio of the arc length to the radius of the circle.
    \(\theta = \frac{{{\text{arc length}}}}{{{\text{radius}}}} = \frac{l}{r}\)
    Therefore arc length \(l=r\theta\)
If we consider the arc to be the total circumference of the circle, then arc length = \(2\pi r\) . Also, we know that the angle subtended at the center of the circle by its circumference is \(360^0\) . Then by the above formula,
    Angle subtended = (arc length)/(radius)
Then,
       \(\theta\) = arc length/radius = s/r radians
Note:

ii) If the measure of an angle in degree and radian be D, G and C respectively, then
\(\frac{D}{{180}} = \frac{G}{{200}} = \frac{C}{\pi }\)