A circle circumscribes a rectangle whose sides are in the ratio of $4:3$. If the area of the rectangle is $192{\text{ cm}}^2$, then the perimeter (in $\text{cm}$) of the circle is:

Let the sides of the rectangle be $4x$ and $3x$.Given that the area of the rectangle is $192{\text{ cm}}^2$:
$$\text{Area}=\text{length}\times \text{breadth}$$
$$4x\times 3x=192$$
$$12x^2=192⟹x^2=16⟹x=4$$
So, the dimensions of the rectangle are:
$$\text{Length}=4(4)=16\text{ cm}$$
$$\text{Breadth}=3(4)=12\text{ cm}$$
A circle circumscribing a rectangle has its diagonal equal to the diameter of the circle.
$$\text{Diagonal}=\sqrt{{\text{length}}^2+{\text{breadth}}^2}=\sqrt{16^2+12^2}=\sqrt{256+144}=\sqrt{400}=20\text{ cm}$$
Thus, the diameter of the circle $d=20\text{ cm}$, which implies the radius $r=10\text{ cm}$.The perimeter (circumference) of the circle is:
$$\text{Perimeter}=2\pi r=2\pi (10)=20\pi \text{ cm}$$