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Hint: First use circumference of the circle formula $2 \pi r$ to find the radius of the circle. Later divide the area of the circle formula by 4 to find the area of the quadrant since the quadrant of a circle is ${\dfrac{1}{4}^{th}}$ of the circle.

Complete step-by-step answer:

Given, Circumference of the circle = 22 cm

As we know that Circumference of the circle with radius ‘r’ is $2 \pi r$

Taking $\pi = \dfrac{{22}}{7} $

$\Rightarrow 2\pi r = 22 \Rightarrow r = \dfrac{{22}}{{2\pi }} = \dfrac{{22 \times 7}}{{2 \times 22}} = 3.5cm$

Since, quadrant of a circle means ${\dfrac{1}{4}^{th}}$ of the circle

$ \therefore$ Area of quadrant of a circle =$ \dfrac{{{\text{Area of the circle}}}}{4}{\text{ = }}\dfrac{{\pi {r^2}}}{4}{\text{ = }}\dfrac{{22 \times {{\left( {3.5} \right)}^2}}}{{7 \times 4}} = 9.625{\text{ }}c{m^2} $

Note - In these types of problems, a common parameter(which is radius here) between the given data(circumference) and the data which needs to be computed(area) is to be found using some formulas.