why should we define "the magnetic flux is the dot product of the magnetic field and the plane area , instead of perimeter of the plane
"ϕB  is the dot product of the magnetic field B and the plane area a"


No, it's: $$ \Phi_A = \iint_A\vec{B}\cdot d\vec{A}$$ where B and dA are both vectors. You can't have a dot product between scalars.

B is a vector that points to the direction of the field and its magnitude is the field strength.

dA is the normal vector of the surface infinitesimal (an infinitely small part of the surface). That is, it points perpendicularly away from the surface and its magnitude is the area of the surface.

The dot product gives you the magnitude of the component of one vector that is parallel to the the other vector multiplied by the magnitude of the other vector.

Flux is the amount of magnetic field that is "flowing" through a given surface. So you only want to calculate with the component of B that is perpendicular to the surface. Hence the dot product.

If B is uniform in magnitude and is perpendicular to the surface at each point than the equation simplifies to just $$\Phi_A=B*A$$ where B is the magnitude of B and A is the total are of the surface. Because then B gives you the "flow rate" per unit area.

If B is uniform in magnitude but has a uniform angle with the surface at each point, than you have to take the part of B that is "flowing" through the surface, the perpendicular component that is. That's why you need the dot product. In this case the equation is $$\Phi_A=\vec{B}\cdot\vec{A}=B_\perp*A$$


  "instead of perimeter ,p , of the plane"


Why would you possibly want to multiply by the perimeter? That is just simply not what flux is.