Advanced Geometry by H.C. Rajpoot-2014
An infinitely long vertical cylinder, with a diameter d, vertically stands on the horizontal plane. Find out the solid angle (in sr) subtended by the cylinder at a point lying on the same plane at a distance 7d from the longitudinal axis.
Note: above picture is given for visualization.
The answer is 0.143.
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1 solution
The angle subtended by the cylinder as seen from the point is
x = 2 A r c S i n ( 7 d 2 1 d ) = 2 A r c S i n ( 1 4 1 )
This angle x is 2 π x of the circumference, while the hemisphere has a solid angle of 2 π . Therefore, the infinitely long vertical cylinder subtends a solid angle of
2 S i n ( 1 4 1 ) = 0 . 1 4 2 9 7 9 . . .
The cylinder at infinity appears to be a point at the upper pole of the hemisphere.