One side of a rectangle is 12 cm long and its diagonal measures 37 cm.

One side of the rectangle = 12 cm
Diagonal of the rectangle = 37 cm

The diagonal of a rectangle forms the hypotenuse of a right-angled triangle. The other two sides of the triangle are the length and the breadth of the rectangle.

Now, using Pythagoras' theorem, we have:

(one side) $^{2}+(\text { other side })^{2}=$ (hypotenuse) $^{2}$

$\Rightarrow(12)^{2}+(\text { other side })^{2}=(37)^{2}$

$\Rightarrow 144+(\text { other side })^{2}=1369$

$\Rightarrow(\text { other side })^{2}=1369-144$

$\Rightarrow(\text { other side })^{2}=1225$

$\Rightarrow$ other side $=\sqrt{1225}$

$\Rightarrow$ other side $=35 \mathrm{~cm}$

Thus, we have:
Length  = 35 cm
Breadth = 12 cm

Area of the rectangle $=35 \times 12=420 \mathrm{~cm}^{2}$