This is the most general case for any location of given point in the space which is derived by using basic formula taken from the book "Advanced Geometry by H.C. Rajpoot". Its derivation & detailed explanation has been given in author's book of research articles of 3-D Geometry.

1. Solid angle subtended by a rectangular plane at any point in the space
*Mr Harish Chandra Rajpoot (B Tech, ME)* 12 sept, 2013
Madan Mohan Malaviya University of Technology, Gorakhpur-273010 (UP) India
Let there be a rectangular plane ABCD having length ‘l’ & width ‘b’ ( ) and a given point say P (observer) at a distance ‘r’ from the centre O of the plane
(as shown in the figure below)
Fig: Solid Angle subtended by a rectangular plane at any point P in the space
Now, draw a perpendicular PQ from the given point ‘P’ to the plane of rectangle ABCD & join the given point ‘P’ & the foot of perpendicular ‘Q’ to the centre ‘O’ such that
is the angle of inclination of the line OP with OQ (or with the given plane ABCD)
(Also called ‘angle of elevation’ of the given point ‘P’ in the space)
is the angle between the line OQ & the reference line*
(Also called ‘angle of deviation’ of the given point ‘P’ in the space)
(*Reference line: The line passing through the centre & parallel to the longer side (i.e. AB & CD) of the given rectangular plane. )
Now, extend the sides AB & CD and draw the lines QK & FE passing through the point ‘Q’ & parallel to the sides AB & BC respectively. Extended lines AB, CD, QK & the reference line intersecting the line FE at the points ‘E’, ‘F’, ‘Q’ & ‘L’ respectively.
In right

2. ⇒ ⇒ ⇒
Where PQ is the normal height of the given point ‘P’ from the rectangular plane ABCD
In right ⇒ ⇒ ⇒ ⇒ ⇒ ⇒ ⇒
It is clear from the above figure that the solid angle subtended by the rectangular plane ABCD at the given point P lying on the axis PQ at a normal height ⇒
Now, for ease of calculation let’s assume
&
Here, we would directly use the formula for solid angle subtended by a rectangular plane of size at any point lying at a normal height h from any of the vertices given as follows ( √( )( ) )
*Above result is directly taken from the book which has its derivation & explanation in details.

3. { √( )( ) } { √( )( ) } { √( )( ) } { √( )( ) }
Now, on setting the corresponding values we can find the solid angle subtended by the given rectangular plane ABCD at the point ‘P’ as follows
…………………….. (1)
We find that the value of depends on the following variables
Case 1: The solid angle subtended by the square plane having each side of length ‘a’ at any point in the space can be obtained by putting l = b = a in the above expressions of eq(1) as follows
& and { √( )( ) } { √( )( ) } { √( )( ) } { √( )( ) }
Now, on setting the corresponding values we can find the solid angle subtended by the given square plane at the given point as follows
Case 2: The solid angle subtended by the rectangular plane at any point lying on the axis normal to the plane & passing through the centre ‘O’ is obtained by setting in the above expressions of eq(1) as follows

4. & and { ( )( ) √(( ) )(( ) ) } { ( )( ) √(( ) )(( ) ) } { ( )( ) √(( ) )(( ) ) } { ( )( ) √(( ) )(( ) ) } ⇒ { √( )( ) } { √( )( ) } { √( )( ) } { √( )( ) }
Now, on setting the corresponding values we can find the solid angle subtended by the given rectangular plane at any point lying on the axis normal to the plane & passing through the centre as follows ⇒ { √( )( ) } { √( )( ) } { √( )( ) } { √( )( ) } ⇒ { √( )( ) }

5. Case 3: The solid angle subtended by the rectangular plane at a point lying on the centre of the plane is obtained by setting in the above expressions of the eq(1) as follows ( ) ( ) ( ) ( )
& ( ) height and { ( )( ) √(( ) ( ) )(( ) ( ) ) } { ( )( ) √(( ) ( ) )(( ) ( ) ) } { ( )( ) √(( ) ( ) )(( ) ( ) ) } { ( )( ) √(( ) ( ) )(( ) ( ) ) } { } * + { } * + { } * + { } * +
Now, on setting the corresponding values we can find the solid angle subtended by the given rectangular plane at a point lying on the centre of the plane as follows ⇒
*It’s also true for any point lying on the plane inside the boundary of rectangular plane.
Note: Above results have been taken from the book “Advanced Geometry by Harish Chandra Rajpoot” copyrighted by the Notion Press publication Chennai, India in December, 2013. ISBN-13: 9789383808151, ISBN-10: 9383808152 (www.notionpress.com)