CONCEPT PREVIEW Find the area of each sector.

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Identify the formula for the area of a sector: \( A = \frac{1}{2} r^2 \theta \), where \( r \) is the radius and \( \theta \) is the angle in radians.

Ensure the angle \( \theta \) is in radians. If it's given in degrees, convert it using \( \theta_{\text{radians}} = \theta_{\text{degrees}} \times \frac{\pi}{180} \).

Substitute the given radius \( r \) and the angle \( \theta \) in radians into the formula.

Calculate the expression \( \frac{1}{2} r^2 \theta \) to find the area of the sector.

Review the units of your answer to ensure they are in square units, consistent with the units of the radius.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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 calculating the area of the sector, as the area is directly proportional to this angle. The central angle can be expressed in degrees or radians, with 360 degrees or 2π radians representing a full circle.

Area Calculation

Calculating the area of a sector involves understanding the relationship between the central angle and the total area of the circle. The formula A = (θ/360) * πr² allows for the determination of the sector's area based on its angle and radius. This concept is fundamental in trigonometry and geometry, linking circular measurements to area.

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