I randomly deploy sensors of radius 10 cm and of circular sensing range in the square of dimensions 50*50 cm. I want to find the overlapped area between the sensing range
Google search for overlap on three circles gave a 27 page document from Australian DoD... Honestly, I'm not sure it's worth deriving an expression from trig principles for this.
If generalization isn't needed, you could use a CAD program to manually calculate this. Draftsight is free and my favorite. You could draw circles (type Circle), then use the Trim command to cut away the intersection, make it a polyline using PEdit, Multiple, Join, then use GetArea to calculate the area of the intersection. Just based on experience (haven't tried it), that should work unless there's some weird problem with the polyline conversion. If you're not familiar with the software, there's YouTube videos on the commands and Draftsight in general.
Find the intersection points between the circles. It is easy to do with the circle equation. Then use the radius and these point to calculate the angle to the center of the circle. Then use the angle to create the arc equation. integrate this equation between these two nodes. Multiple the answer by two. This is the area.