Finding the Area of a Circle Sector Explained

The question asks us to find the area of a sector of a circle given its radius and the central angle. A sector of a circle is the portion of a disk enclosed by two radii and an arc. Its area is a fraction of the entire circle's area, determined by the ratio of the sector's central angle to the total angle in a circle (360 degrees or \(2\pi\) radians).

Formula for Area of a Sector

The area of a sector can be calculated using the formula:

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

Where:

• \(\theta\) is the central angle of the sector in degrees.
• \(r\) is the radius of the circle.
• \(\pi\) is a mathematical constant, approximately 3.14159.

Applying the Formula to the Given Problem

We are given:

• Radius, \(r = 5 \text{ cm}\)
• Central angle, \(\theta = 60^\circ\)

Substitute these values into the formula:

$$ \text{Area} = \frac{60^\circ}{360^\circ} \times \pi (5 \text{ cm})^2 $$

First, simplify the fraction representing the ratio of the angle to the full circle:

$$ \frac{60^\circ}{360^\circ} = \frac{1}{6} $$

Next, calculate the square of the radius:

$$ (5 \text{ cm})^2 = 25 \text{ cm}^2 $$

Now, substitute these simplified values back into the area formula:

$$ \text{Area} = \frac{1}{6} \times \pi \times 25 \text{ cm}^2 $$

$$ \text{Area} = \frac{25\pi}{6} \text{ cm}^2 $$

To get a numerical value, use the value of \(\pi\) (approximately 3.14159):

$$ \text{Area} \approx \frac{25 \times 3.14159}{6} \text{ cm}^2 $$

$$ \text{Area} \approx \frac{78.53975}{6} \text{ cm}^2 $$

$$ \text{Area} \approx 13.089958... \text{ cm}^2 $$

Rounding the Result

The question asks for the area rounded off to one decimal place. Rounding 13.089958... to one decimal place gives 13.1 \text{ cm}^2.

Comparison with Options

Let's compare our calculated area with the given options:

1. 12.8 cm²
2. 14.1 cm²
3. 13.1 cm²
4. 15.1 cm²

Our calculated area, rounded to one decimal place, is 13.1 cm², which matches option 3.

Conclusion

Using the formula for the area of a sector, with a radius of 5 cm and a central angle of 60°, the calculated area is approximately 13.09 cm². When rounded to one decimal place, this becomes 13.1 cm².

Revision Table: Key Concepts for Area of Sector

Additional Information: Units and Angles

It is crucial to pay attention to the units of the angle when using the area of sector formula. The formula \(\frac{\theta}{360^\circ} \times \pi r^2\) requires the angle \(\theta\) to be in degrees. If the angle is given in radians, the formula becomes \(\frac{\theta_{rad}}{2\pi} \times \pi r^2\), which simplifies to \(\frac{1}{2} r^2 \theta_{rad}\).

In this problem, the angle was given in degrees (60°), so we correctly used the formula involving 360°. The unit for the area is always in square units, matching the unit of the radius (e.g., cm², m², in²).