In a circle of radius 7 cm, an arc subtends an angle of 108° at the centre. The area of the sector is :

Question

Accepted Answer

To determine the area of the sector formed by an arc that subtends an angle of 108° at the center of a circle with a radius of 7 cm, we can use the formula for the area of a sector:

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

where $\theta$ is the angle in degrees, and $r$ is the radius of the circle.

Given:
- $\theta = 108^\circ$
- $r = 7 \text{ cm}$

Substituting these values into the formula, we get:

$$
\text{Area of sector} = \frac{108}{360} \times \pi \times (7)^2
$$

Simplifying the fraction:

$$
\frac{108}{360} = \frac{3}{10}
$$

Now, calculate the area:

$$
\text{Area of sector} = \frac{3}{10} \times \pi \times 49
$$

$$
= \frac{147}{10} \times \pi
$$

Using $\pi \approx 3.14$:

$$
= \frac{147}{10} \times 3.14
$$

$$
= 14.7 \times 3.14
$$

$$
= 46.158 \text{ cm}^2
$$

Rounding to one decimal place, the area of the sector is approximately $46.2 \text{ cm}^2$.

Therefore, Option 4: $46.2 \text{ cm}^2$ is the correct answer.

**Explanation for Incorrect Options:**

1. **Option 1: $43.2 \text{ cm}^2$** - This value is less than the calculated area of $46.2 \text{ cm}^2$. It does not match the result obtained from the formula and calculations.

2. **Option 2: $44.2 \text{ cm}^2$** - This value is also less than the calculated area. It is not consistent with the correct calculation of the sector's area.

3. **Option 3: $45.2 \text{ cm}^2$** - While closer to the correct answer than Options 1 and 2, this value is still less than the calculated area of $46.2 \text{ cm}^2$ and does not match the precise calculation.

In conclusion, Option 4 is the correct answer as it accurately reflects the calculated area of the sector based on the given parameters.