Connection of the radian with other units (minutes and seconds)

 

An angle of 1 radian

 

The proportion of the radian and other units of angular measurement is described by the formula:

1 rad = 1/2π turns = 180°/π degrees = 200g/π grads.

Obviously, a flat angle equals 180° or (π·R)/R = π rad.

This suggests a trivial formula of translation from degrees, minutes, and seconds to radians and vice versa.

Radians to degrees. Degrees to radians

where α[rad] is an angle measured in radians, and [°] is an angle measured in degrees.

The value of 1 radian

More specifically:

The value of 1 radian

1 radian

The value of 1 radian

The value of 1 radian in minutes

 

1 radian in minutes

More specifically:

1 radian in minutes

The value of 1 radian in seconds

1 radian in seconds

More specifically:

1 radian in seconds

In the metric system of angular measurements, a right angle is divided into 100 grads, and each grad is divided into 100 centigrads, which in turn is divided into hundredths of a centigrad, so r” (or 1 rad in hundredths of centigrad) = (400·100·100)/2 π = 636620.

 

Centigrad (1 rad in hundredths) = 636620

 

This is hardly used at all, since the metric system of angular measurements has not yet found wide application.

When converting radians to degrees (or minutes, or seconds), we turn an abstract number (rad) into a named one (r°,r’,r’’), so we have to multiply by r° or r’,r’’

But when converting degrees to radians, we eliminate the naming: we obtain an abstract number; so, we have to divide by r° or r’,r’’ or multiply by a reciprocal 1/r° or 1/r’,1/r’’.