I think you mean

and also you want

.

Supposed the trajectory is angled

horizontally from the vertical plane through the

*y*-axis, taking

to be positive to the right and negative to the left of the

*y*-axis. The straight line on the ground through the origin making this angle with the

*y*-axis intersects the circle at two points. The distances from the origin to these two points can be calculated; these are the minimum and maximum ranges respectively at the angle

. Let

be the vertical angle from the ground at which the particle must be projected to cover the minimum range, and

be the vertical angle to reach the maximum range. (This can be computed from the formula

where

is the acceleration due to gravity and

the desired range.) The maximum value of the horizontal angle

is

.

Then the solid angle you want is: