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"No" it is has only unit (steradian) not DF

Approach=

Step 1. formula-   solid angle is the ratio of area of segment to the square of radius of the sphere of which  the segment  is a part

Step 2. DF of each paramter in the formula, as area and square of radius both are a type of length square or in other words "both" have dimension of length square so their ratio becomes dimensionless.

Following notations / inferences (right side) may be written / considered when you read the lines below (left side)

Uniform linear motion                                      a=0

started from rest                                          u=0

comes to rest                                              v=0

stopped                                                    v=0

pentrated through distance x and stopped                   v=0 

straight line and unidirectional                 speed=velocity (magnitude)

vertically upward motion                         u and g opposite sign

vertically downward motion                       u and g same sign

higher slope of x-t graph                        higher velocity

displacement in 5th second                       dispacement in 5 second - displacement in 4 second

constant velocity                                graph between v & t  will be a line paralled to time axis. 

Area under v-t graph                             Displacement as x=vt

Best Wishes

Following step by step method may be followed for solving NEW physics numericals:

Step 1-  P- M i.e. Physics to maths, write given values with thier NOTATION

e.g. if acceleration is given as 5 meter per second sqaure then wrtite a=5 ms-2 , time 2 sec as t=2s

Step-2 H-M i.e hidden to maths, write hints and indirect statement by identifying correct notation

e.g. if it is given that body was initialy at rest or started from rest then write u=0

Please note- Convert all given values in MKS system (or any other single system of units) before calculation

Step-3 TF/DF/CF, topic formula/desired formula/chapter formula

now decide which formula is to be utilised for desired quantity like if final velocity v is asked then look for formula "having" v,u,a and t (given and asked) 

if you are unable to pick suitable formula then write chapter formula(main formula of a chapter) then look how can you find the quantities involved in the formula, for example if change in momentum is asked then find mass(m) using given data (mass may be given by the hint that force of 10 N produces accelration of 2 m per second sqaure then it will simply mean that mass is 5 kg)
 

In above case two formulas are utilised so correlation is done in step-4 

Step-5 put values and solve neatly.

Please note- Initially some students may find it boring and lenthy but with passage of time and practice it will become your habit to approach all questions by this method and then it will take quite less time to analyze and solve problems even without writing many of the steps. Once it becomes your habit, it will become quite easier to approach new questions which you have not solved earlier.

Best wishes