An arc of length $121\text{cm}$ makes an angle $77^{\circ }$ at the centre of a circle. The radius of the circle is:

The formula for arc length $L$ is $L=r\theta$, where $\theta$ is in radians. First, convert $77^{\circ }$ to radians: $\theta =77\times \frac{\pi }{180}=77\times \frac{22}{7\times 180}=\frac{11\times 22}{180}=\frac{242}{180}=\frac{121}{90}$ radians.
Substituting into the formula: $121=r\times \frac{121}{90}$.
Solving for $r$: $r=121\times \frac{90}{121}=90\text{cm}$.