If the difference between the circumference and radius of a circle is 37 cm,

We have given the difference between circumference and radius of the circle.

Let C be the circumference, r be the radius and A be the area of the circle.

Therefore, from the given condition we have

$C-r=2 \pi r-r$

$\therefore 37=2 \pi r-r$

$\therefore 37=r(2 \pi-1)$

$\therefore r=\frac{37}{(2 \pi-1)}$

Now we will substitute $\pi=\frac{22}{7}$.

$\therefore r=\frac{37}{\left(2 \times \frac{22}{7}-1\right)}$

$\therefore r=\frac{37}{\left(\frac{44}{7}-1\right)}$

$\therefore r=\frac{37}{\left(\frac{44-7}{7}\right)}$

$\therefore r=\frac{37}{\left(\frac{37}{7}\right)}$

$\therefore r=37 \times \frac{7}{37}$

$\therefore r=7$

Now we will substitute the value of $r$ in $A=\pi r^{2}$.

$\therefore A=\pi \times 7^{2}$

Now we will substitute $\pi=\frac{22}{7}$.

$\therefore A=\frac{22}{7} \times 7^{2}$

$\therefore A=22 \times 7$

$\therefore A=154$

Therefore, area of the circle is $154 \mathrm{~cm}^{2}$.

Hence, the correct answer is option (a).