Degree

The degree system of measuring angles is a gift to us from the Babylonians (5000-500 BC residents of the present day Iraq). They used a base of 60 for their number system (just like we use a base of 10 for the decimal system and a base of 2 for the hbinary system). They extended this base 60 system to measuring time and angles as well (and we still use both of them).

Corresponding to HMS (Hour-Minutes-Seconds) system of measuring time, there is DMS (Degree- Minutes-Seconds) system of measuring angles. Hence,

1 degree = 60 minutes ('), and
1 minute = 60 seconds (").

Additioally,

1 complete rotation = 360 degrees (probably because Babylonians had 360 days in their calendar year, 30 days in each of 12 months)


Radian

One radian is the angle between two radii of a circle that subscribes an arc of length one radius on the circumference. How do we know how many radians a particular angle is equal to? We simply ask, "How many arcs, each of length one radius, does the angle subscribes on the circumference?" If an angle subscribes an arc of length 2 radii on the circumference, then the angle must be 2 radians.

How many arcs, each one radius long, are there in the whole circumsnference ( a complete rotation)? 2*pi, right? Yeap. So there are 2*pi radians in a complete circle. This gives us:

1 complete rotation = 2*pi radians= 360 degrees
=> pi radians = 180 degrees
=> 1 radian = 57.3 degrees (approximately).


Gradian (gon or grade)

This system of measuring angles divides a complete rotation into 400 gradians or gons or grades. Hence,

1 complete rotation = 400 gradians
=> 1 right angle = 100 gradians
=> 90 degrees = 100 gradians
=> 1 degree = 10/9 gradian.

Equivalently,
1 gradian = 0.9 degree.

Similarly,
2*pi radians = 400 gradians
=> pi radians = 200 gradians
=> 1 radian = 63.662 gradians (approximately).