Let v be the distance from the midpoint to the upper or lower side of the trapezoid and let s be the distance from the midpoint to the sloping sides of the trapezoid.
Let t be the distance from the midpoint (center) of the trapezoid to any point on the sides of the trapezoid.
Assume that theta is zero from the center of the trapezoid to the center of the circle, and increases in a clockwise direction.
Label angle points through the corners of the trapezoid:
UR, upper right : LR, lower right :
LL , lower left : UL upper
and RS as the perpendicular angle to the right side
and LS as the perpendicular angle to the left side.
while theta < UR
t = v/cos(theta)
while UR < theta < LR
t = (s/(cos(theta-RS))
while LT < theta < 180
t = -v/cos(theta)
for theta 180 to 360, it's a mirror image of the above.