The diagram above shows a circular plane, centered at the origin 'O', has a radius $7 cm$. Two identical rectangular strips, each having width $2 cm$, are thoroughly cut off from the circular plane along x & y axes. Thus four identical segments of the circular plane are left over.
Question: How to calculate the solid angle subtended by the remaining plane (consisting of four identical segments) at a point $P (0, 0, 4cm)$ lying on the z-axis (normally outwards to the plane of paper)?
Solid angle $(\omega)$ subtended by the circular plane, with a radius $r$ at any point lying on the geometrical axis at a distance $h$ from the center, is given as $$\omega=2\pi \left(1-\frac{h}{\sqrt{h^2+r^2}}\right)$$ And the solid angle $(\omega)$ subtended by any rectangle of size $l*b$ at any point lying on the perpendicular axis at a distance $d$ from the center is given by standard formula as $$\omega=4sin^{-1}\left(\frac{lb}{\sqrt{(l^2+4d^2)(b^2+4d^2)}}\right)$$
But how to apply these formula on the non radial segments of a circle? Any help is greatly appreciated.
The distance d=4 cm, radius R=7cm and gap g=2cm give a solid angle of 1.70538588 if we use my formula in Solid angle of a Rectangular Plate.