The area of a sector is 22cm. What is the radius of the sector if the angle subtended at the centre is 140 degree

Answer 1

To find the radius of the sector given the area and the angle subtended at the center, you can use the formula for the area of a sector:

$$\text{Area of the sector}=\frac{\theta }{360}\times \pi r^2$$

Where:

• $\theta$ is the angle in degrees,
• $r$ is the radius of the sector.

Given:

• Area of the sector = 22 cm²
• $\theta =140$ degrees

Now, we can rearrange the formula to solve for $r$:

1. Substitute the values into the formula:

$$22=\frac{140}{360}\times \pi r^2$$

2. Simplify $\frac{140}{360}$:

$$\frac{140}{360}=\frac{7}{18}$$

Now we have:

$$22=\frac{7}{18}\times \pi r^2$$

3. To isolate $r^2$, multiply both sides by $\frac{18}{7}$:

$$22\times \frac{18}{7}=\pi r^2$$

4. Calculate the left side:

$$\frac{22\times 18}{7}=\frac{396}{7}\approx 56.5714$$

So:

$$\pi r^2=\frac{396}{7}$$

5. Now, divide both sides by $\pi$ to solve for $r^2$:

$$r^2=\frac{396}{7\pi }$$

6. Finally, take the square root to find $r$:

$$r=\sqrt{\frac{396}{7\pi }}$$

Using $\pi \approx 3.14159$:

$$r\approx \sqrt{\frac{396}{7\times 3.14159}}\approx \sqrt{\frac{396}{21.99193}}\approx \sqrt{18.0009}\approx 4.24\text{ cm}$$

Thus, the radius of the sector is approximately 4.24 cm.