CONCEPT PREVIEW Find the area of each sector.

Verified step by step guidance

1. Step 1: Understand the concept of a sector. A sector is a portion of a circle, resembling a 'slice of pie', defined by two radii and the arc between them.
2. Step 2: Recall the formula for the area of a sector: \( A = \frac{1}{2} r^2 \theta \), where \( r \) is the radius of the circle and \( \theta \) is the central angle in radians.
3. Step 3: If the angle is given in degrees, convert it to radians using the conversion \( \theta_{\text{radians}} = \theta_{\text{degrees}} \times \frac{\pi}{180} \).
4. Step 4: Substitute the values of the radius \( r \) and the angle \( \theta \) (in radians) into the formula.
5. Step 5: Simplify the expression to find the area of the sector.

Key Concepts

Sector of a Circle

A sector of a circle is a portion of the circle enclosed by two radii and the arc between them. It resembles a 'slice' of the circle and is defined by its central angle. The area of a sector can be calculated using the formula A = (θ/360) * πr², where θ is the central angle in degrees and r is the radius.

Central Angle

The central angle of a sector is the angle formed at the center of the circle by the two radii that define the sector. It is crucial for determining the proportion of the circle that the sector represents. The central angle can be expressed in degrees or radians, and it directly influences the area calculation of the sector.

Area Calculation

Calculating the area of a sector involves understanding the relationship between the central angle and the total area of the circle. The total area of a circle is given by A = πr². To find the area of a sector, you take the fraction of the circle represented by the central angle and multiply it by the total area, allowing for precise measurement of the sector's area.