Derivation of Torque on Loop of Wire For a loop of wire in a uniform magnetic field, the torque is given by the following equation:
$\vec{\tau}$ $=$ $\vec{m}$ $\times$ $\vec{B}$
where $\vec{m} = I\vec{A}$ is the magnetic moment of the loop and $\vec{B}$ is the magnetic field. How can you prove this, especially for an arbitrary loop of irregular shape? Why does the torque rely on the area but not the perimeter of the loop?
 A: as your 1st question---
$\vec{\tau}$ $= $ $\vec{m}$ $\times$ $\vec{B}$
well this derivation could be found on any website, just google it you will get it, or  please state what to ask specifically in its derivation
now your 2nd question---
$\vec{m}$ $=$ $I\vec{A}$ is the magnetic moment of the loop and 
is the magnetic field. How can you prove this? Why does the torque rely on the area but not the perimeter of the loop?
if you see the derivation of torque above, the  final solution would come out to be
$\vec{\tau}$ $=$ $IAB\sin\theta$
where $\theta$ is angle by which rectangular loop rotates about its plane from its earlier position 
the above equation is similar to cross product one,  so we can treat area as a vector which is perpendicular to plane of loop(we treated area as vector to calculate current density or electric flux as well soit just fits in) however we know current is scalar
so it could be written in cross product form where area  and magnetic field are vector
---i.e $\vec{\tau}$ $=$ $I\vec{A}\times\vec{B}$ or we can combine vector $I\vec{A}$ as another vector which is magnetic moment $\vec{m}$. 
these derivation are common to find in internet, you will see for yourself that torque would depend on area not perimeter, just search it for rectangular loop for simplicity I hope it helps
