The length of an arc of a circle equivalent to one-fourth of the circle's total length is 22 cm. Find the area of the quadrant.

If the length of an arc is one-fourth of the entire length, then it is a part of a quadrant.

Length of an arc $=\frac{\theta }{{360}^{°}}\times 2\pi r$ 

For a quadrant, $\theta =90°$

Therefore, the length of the arc $=\frac{90}{360}\times 2\pi \text{r}$

$\Rightarrow 22=\frac{90}{360}\times 2\times \frac{22}{7}\times \text{r}$

$\Rightarrow \frac{22\times 360\times 7}{90\times 2\times 22}=r$

$\Rightarrow \text{r}=14$

Therefore the area of the quadrant is:

$=\frac{1}{4}\times \mathrm{\pi }\times {\mathrm{r}}^2$

$=\frac{1}{4}\times \frac{22}{7}\times 14\times 14$

$=154 {\mathrm{cm}}^2$