give derivation that why elecric field is zero at the centre of a polygon.

To get the total you take the vector sum of the individual forces. You put a test charge at O and calculate the force to each of the 5 individual charges.

Each charge is at the same distance and they are separated by an angle of 360/5 = 72º.

force from B.
Angle is 90–72 = 18º
Fx = Kcos 18
Fy = Ksin 18
(K factor includes distance and other constants)

Force from A
Fx = 0
Fy = K

Force from E
Fx = –Kcos18
Fy = Ksin18

Force from D
Fx = –Kcos 54
Fy = –Ksin 54

Force from C
Fx = Kcos54
Fy = –Ksin 54

add them up
Rx = Kcos18 + 0 –Kcos18 –Kcos 54 + Kcos54 = 0
Ry = Ksin18 + K + Ksin18 –Ksin 54 –Ksin 54
Ry = K + 2Ksin18 – 2Ksin54
Ry = k( 1 + 0.618034 – 1.618034) = 0