You can get the area of the curve using differential forms.
In[32]:= tw[t] = TensorWedge[reimz[t], dz[t]][[1, 2]]
Out[32]= ((3 + 25 Cos[t]) (567 + 75 Cos[t] + 250 Sin[t]) (-15000 +
81289 Cos[t] + 55050 Cos[t]^2 + 5625 Cos[t]^3 - 6000 Sin[t] +
183500 Cos[t] Sin[t] + 37500 Cos[t]^2 Sin[t] +
62500 Cos[t] Sin[t]^2))/(16 (367 + 75 Cos[t] +
250 Sin[t])^3) + (5 (334 + 150 Cos[t] + 1335 Sin[t] +
375 Cos[t] Sin[t] + 1250 Sin[t]^2) (6000 Cos[t] +
50000 Cos[t]^2 + 206289 Sin[t] + 55050 Cos[t] Sin[t] +
5625 Cos[t]^2 Sin[t] + 233500 Sin[t]^2 +
37500 Cos[t] Sin[t]^2 + 62500 Sin[t]^3))/(16 (367 + 75 Cos[t] +
250 Sin[t])^3)
In[33]:= 1/2 NIntegrate[tw[t], {t, 0, 2 \[Pi]}]
Out[33]= 1.95895