The length of an arc is 90cm , the angle subtended at the center of the circle is 240 degree . Calculate the radius of the circle

The length of an arc ($L$) is given by:

$$L = \frac{\theta}{360^\circ} \times 2\pi r$$

where $\theta$ is the central angle in degrees and $r$ is the radius.

We are given $L = 90$ cm and $\theta = 240^\circ$.

$$90 = \frac{240^\circ}{360^\circ} \times 2\pi r$$

$$90 = \frac{2}{3} \times 2\pi r$$ $$90 = \frac{4\pi r}{3}$$

Multiply both sides by 3:

$$90 \times 3 = 4\pi r$$ $$270 = 4\pi r$$

Divide both sides by $4\pi$:

$$r = \frac{270}{4\pi}$$ $$r = \frac{135}{2\pi} \text{ cm}$$

Using $\pi \approx 3.14159$:

$$r \approx \frac{135}{2 \times 3.14159}$$ $$r \approx \frac{135}{6.28318}$$ $$r \approx 21.48 \text{ cm}$$

Final Answer: The radius of the circle is $\boxed{\frac{135}{2\pi} \text{ cm}}$ or approximately $\boxed{21.48 \text{ cm}}$.