The hydraulic radius is variable used in open channel flow for the Manning Equation and the Hazen Williams Equation.
In general, the Hydraulic Radius of a system can be solved by: R = \frac{\text{crosssectional flow area}}{\text{wetted perimeter}} = \frac{A}{s}

where:
 R = the Hydraulic Radius
 wetted perimeter = perimeter where water touches a solid surface
Hydraulic Radius for Common Channel Shapes:
The following common channel/pipe shapes have been simplified to easily find their hydraulic radius'.
Rectangular Channel (partially full):
For a rectangular Channel, given h depth and b width:
Circle (half full):
For a half filled circle, given r radius & D diameter:
Circle (full):
For a full circle, given r radius & D diameter:
Square (full):
For a full square, given s side:
Resources:
 Stephen Elmer Slocum, "Elements of Hydraulics", 1915 (Read Online Here)