CONCEPT PREVIEW Find the area of each sector.

Verified step by step guidance

1. Identify the given information for the sector: the radius \(r\) of the circle and the central angle \(\theta\) of the sector. The angle should be in degrees or radians.
2. Recall the formula for the area of a sector: \(\text{Area} = \frac{\theta}{360} \times \pi r^{2}\) if \(\theta\) is in degrees, or \(\text{Area} = \frac{1}{2} r^{2} \theta\) if \(\theta\) is in radians.
3. If the angle \(\theta\) is given in degrees, use the first formula. If it is in radians, use the second formula. Convert the angle to the appropriate unit if necessary.
4. Substitute the values of \(r\) and \(\theta\) into the chosen formula to set up the expression for the area of the sector.
5. Simplify the expression to find the area of the sector (do not calculate the final numeric value unless asked).

Key Concepts

Definition of a Sector

A sector of a circle is a portion bounded by two radii and the arc between them. It resembles a 'slice' of the circle, and its size depends on the central angle that subtends the arc.

Formula for Area of a Sector

The area of a sector is given by (θ/360) × πr² when θ is in degrees, where r is the radius of the circle. This formula calculates the fraction of the circle's area corresponding to the central angle.

Conversion Between Radians and Degrees

Angles can be measured in degrees or radians. Since the sector area formula depends on the angle unit, converting between radians and degrees (1 radian = 180/π degrees) is essential for correct calculations.